1 Abstract

The muHVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.
Data Projection: Dimension projection of the compressed cells to 1D,2D or 3D with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.
Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.
Prediction: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

Row.No	price	speed	hd	ram	screen	ads
1	-1.2973	-1.1954	-1.3136	-0.7176	-0.615	-2.388
2	-0.7999	-0.7834	-1.2898	-1.1078	-0.615	-2.388
3	-1.1360	-1.1954	-0.8854	-0.7176	0.549	-2.388
4	-0.7091	-1.1954	-0.8854	0.0630	-0.615	-2.388
5	1.7210	-0.7834	-0.0767	1.6240	-0.615	-2.388
6	2.3932	0.9159	-0.0767	1.6240	-0.615	-2.388

3 Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Flow map with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

Hierarchical Vector Quantization using k−means
Hierarchical Vector Quantization using k−medoids

For more information on vector quantization, refer the following link.

The HVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the HVT function first before moving ahead.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters have been explained below :

dataset - A dataframe with numeric columns.
min_compression_perc - An integer indicating the minimum percent compression rate to be achieved for the dataset.
n_cells - An integer indicating the number of cells per hierarchy (level).
depth - An integer indicating the number of levels. (1 = No hierarchy, 2 = 2 levels, etc …).
quant.error - A number indicating the quantization error threshold. A cell will only breakdown. into further cells if the quantization error of the cell is above the defined quantization error threshold.
distance_metric - The distance metric can be L1_Norm or L2_Norm. L1_Norm is selected by default. The distance metric is used to calculate the distance between an n dimensional point and centroid. The user can also pass a custom function to calculate this distance.
error_metric - The error metric can be mean or max. max is selected by default. max will return the max of m values and mean will take mean of m values where each value is a distance between a point and centroid of the cell. Moreover, the user can also pass a custom function to calculate the error metric.
quant_method - The quantization method can be kmeans or kmedoids. kmeans is selected by default.
normalize - A logical value indicating whether the columns in your dataset need to be normalized. Default value is TRUE. The algorithm supports Z-score normalization.
diagnose - A logical value indicating whether user wants to perform diagnostics on the model. Default value is TRUE.
hvt_validation - A logical value indicating whether user wants to holdout a validation set and find mean absolute deviation of the validation points from the centroid. Default value is FALSE.
train_validation_split_ratio - A numeric value indicating train validation split ratio. This argument is only used when hvt_validation has been set to TRUE. Default value for the argument is 0.8.

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce the level of detail captured in the data representation

First, we will construct map A by using the below mentioned model parameters.

We will pass the below mentioned model parameters along with training dataset to HVT function.

Model Parameters

Number of Cells at each Level = 200
Maximum Depth = 1
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

set.seed(240)
map_A <- list()
map_A  <- muHVT::HVT(
  trainComputers,
  n_cells = 200,
  depth = 1,
  quant.err = 0.2,
  projection.scale = 10,
  normalize = F,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary.

compressionSummaryTable(map_A[[3]]$compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	200	82	0.41	n_cells: 200 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 41% of cells have reached the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

Number of Cells at each Level = 450
Maximum Depth = 1
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

map_A <- list()
map_A <-muHVT::HVT(trainComputers,
                n_cells = 450,
                quant.err = 0.2,
                depth = 1,
                distance_metric = "L1_Norm",
                error_metric = "max",
                quant_method = "kmeans",
                normalize = F
)

As per the manual, map_A[[3]] gives us detailed information about the hierarchical vector quantized data. map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A

summaryTable(map_A[[3]]$summary,scroll = T,limit = 500)

Segment.Level	Segment.Parent	Segment.Child	n	Cell.ID	Quant.Error	price	speed	hd	ram	screen	ads
1	1	1	9	316	0.25	0.72	0.92	-0.32	-0.02	0.55	-1.15
1	1	2	9	147	0.13	-1.19	-0.78	0.06	-0.72	0.55	-0.49
1	1	3	8	67	0.05	-1.06	-1.20	-0.70	-0.72	-0.61	0.94
1	1	4	6	304	0.16	-0.47	0.92	0.24	0.06	0.55	-1.30
1	1	5	12	188	0.16	-0.93	0.92	-0.01	-0.72	-0.61	0.43
1	1	6	14	109	0.09	-1.50	-0.78	-0.08	-0.72	-0.61	-0.09
1	1	7	10	89	0.14	0.18	0.92	-1.05	-0.72	-0.61	-2.29
1	1	8	7	286	0.11	0.43	0.09	0.49	0.06	-0.61	-0.56
1	1	9	11	11	0.14	-1.49	-1.05	-1.18	-1.00	-0.61	1.52
1	1	10	10	390	0.24	0.45	0.96	0.72	1.62	0.55	-0.70
1	1	11	5	367	0.04	0.56	0.92	0.82	1.62	-0.61	-0.30
1	1	12	9	382	0.09	0.78	0.92	0.82	1.62	-0.61	1.29
1	1	13	8	78	0.17	-1.40	-0.78	0.24	-0.72	-0.61	-1.19
1	1	14	15	111	0.07	-0.61	-1.20	-0.68	-0.72	-0.61	0.14
1	1	15	6	307	0.21	-0.01	2.67	-0.08	-0.20	-0.61	0.92
1	1	16	9	63	0.06	-1.27	-0.78	-0.97	-0.72	-0.61	0.93
1	1	17	9	349	0.11	0.17	2.67	0.26	0.06	-0.61	1.52
1	1	18	7	322	0.27	-0.11	-0.84	0.46	-0.27	2.88	0.13
1	1	19	10	432	0.06	1.35	-0.78	3.06	3.19	-0.61	-0.11
1	1	20	15	424	0.32	1.22	2.67	0.78	1.62	0.55	1.11
1	1	21	13	248	0.12	-0.03	-0.82	0.11	0.06	0.55	0.15
1	1	22	5	331	0.14	-0.45	0.92	-0.20	-0.72	2.88	-0.64
1	1	23	10	151	0.05	0.00	-0.78	-0.68	-0.72	-0.61	0.14
1	1	24	13	374	0.06	1.28	0.09	0.84	1.62	-0.61	0.76
1	1	25	5	380	0.24	2.20	0.92	0.51	-0.25	0.55	-0.80
1	1	26	11	50	0.08	-1.65	-1.20	-0.66	-0.72	-0.61	0.32
1	1	27	14	57	0.13	-1.08	-1.20	-0.82	-0.72	0.55	0.73
1	1	28	14	345	0.26	1.21	0.92	0.58	0.01	0.55	0.06
1	1	29	19	394	0.08	1.44	0.92	0.83	1.62	-0.61	0.72
1	1	30	13	123	0.08	-0.45	0.09	-1.19	-1.11	-0.61	0.10
1	1	31	20	396	0.15	0.28	-0.78	1.73	1.62	0.55	-0.64
1	1	32	6	90	0.18	-0.68	-0.85	-0.66	-0.72	0.55	1.52
1	1	33	11	74	0.08	-0.62	-0.82	-1.01	-0.72	-0.61	-1.08
1	1	34	6	372	0.13	1.76	0.92	0.82	0.06	0.55	0.84
1	1	35	15	337	0.07	0.66	-0.78	0.82	1.62	-0.61	0.31
1	1	36	12	220	0.13	-0.80	-0.78	0.19	0.06	0.55	0.51
1	1	37	11	305	0.19	1.18	0.92	-0.46	-0.72	0.55	0.32
1	1	38	6	71	0.06	-1.65	-0.78	-0.69	-0.72	-0.61	0.25
1	1	39	18	436	0.19	1.33	-0.78	3.06	3.19	0.55	-0.38
1	1	40	24	267	0.08	-0.34	0.92	0.32	0.06	-0.61	-0.16
1	1	41	9	293	0.08	0.10	0.92	0.39	0.06	-0.61	0.99
1	1	42	19	49	0.06	-0.99	-0.78	-1.19	-1.11	-0.61	0.75
1	1	43	6	273	0.13	0.42	-0.78	0.77	0.06	-0.61	-1.00
1	1	44	3	262	0.05	1.21	-0.78	-0.08	0.06	-0.61	0.50
1	1	45	9	371	0.18	2.17	0.92	0.77	-0.72	0.55	0.38
1	1	46	5	117	0.05	-0.37	-0.78	-1.08	-0.72	-0.61	0.60
1	1	47	5	52	0.02	-1.55	-0.78	-1.08	-0.72	-0.61	0.06
1	1	48	9	234	0.11	-0.34	-0.97	0.83	0.06	-0.61	0.46
1	1	49	5	413	0.18	1.12	0.92	-0.42	0.06	2.88	-1.19
1	1	50	27	213	0.12	0.28	0.09	-0.59	0.06	-0.61	-2.29
1	1	51	5	421	0.2	0.29	0.92	2.04	0.06	2.88	-0.71
1	1	52	22	419	0.42	1.00	2.67	1.09	1.62	-0.40	-0.09
1	1	53	16	407	0.36	1.34	0.56	-0.25	1.62	-0.18	-2.26
1	1	54	4	193	0.1	-1.14	0.09	-0.08	0.06	-0.61	-0.42
1	1	55	11	134	0.07	-0.42	-0.78	-0.59	-0.72	-0.61	-0.41
1	1	56	8	86	0.05	-0.92	-0.78	-1.12	-0.72	-0.61	0.45
1	1	57	13	142	0.11	-1.00	-0.78	0.23	-0.72	-0.61	-0.34
1	1	58	19	84	0.08	-1.04	-1.20	-0.69	-0.72	-0.61	0.35
1	1	59	7	12	0.1	-1.46	-1.20	-1.19	-1.11	0.55	0.63
1	1	60	10	437	0.08	1.53	0.92	3.06	3.19	-0.61	-0.11
1	1	61	16	388	0.1	1.40	0.92	0.71	1.62	-0.61	0.13
1	1	62	4	338	0.16	1.85	-0.13	0.45	0.06	0.55	0.26
1	1	63	10	239	0.16	-0.54	0.92	-0.04	-0.72	0.55	0.50
1	1	64	9	224	0.04	-0.15	-0.78	0.33	0.06	-0.61	0.73
1	1	65	21	384	0.17	1.26	0.92	-0.07	1.62	0.55	0.40
1	1	66	11	393	0.21	0.23	0.92	0.67	0.06	2.88	-0.12
1	1	67	8	333	0.18	-0.29	0.92	1.78	0.06	0.40	0.07
1	1	68	22	199	0.2	0.24	-0.78	-0.64	-0.04	0.55	-2.31
1	1	69	9	163	0.12	-0.38	-0.78	-0.57	0.06	-0.61	-1.41
1	1	70	9	246	0.14	-0.65	0.92	0.00	0.06	-0.61	0.15
1	1	71	11	48	0.06	-1.14	-1.20	-1.13	-0.72	-0.61	0.72
1	1	72	5	326	0.19	0.91	0.59	0.84	0.06	-0.61	-1.08
1	1	73	8	214	0.1	0.25	-0.89	-0.09	0.06	-0.61	-1.67
1	1	74	18	181	0.14	0.17	0.09	-0.67	-0.72	-0.61	-0.02
1	1	75	10	2	0.07	-1.14	-0.78	-1.29	-1.11	-0.61	-2.28
1	1	76	15	21	0.15	-1.31	-1.06	-1.12	-0.87	-0.61	-1.08
1	1	77	3	450	0.04	5.26	0.92	4.01	4.75	2.88	0.46
1	1	78	15	83	0.06	-1.19	-0.78	-0.67	-0.72	-0.61	0.90
1	1	79	9	105	0.11	-1.23	0.09	-0.88	-0.72	-0.61	0.81
1	1	80	11	297	0.13	0.88	0.92	-0.15	0.06	-0.61	0.10
1	1	81	5	106	0.02	-0.79	-0.78	-0.89	-0.72	-0.61	0.07
1	1	82	23	420	0.39	2.33	0.92	0.30	-0.04	2.88	0.60
1	1	83	21	320	0.23	0.16	0.92	0.87	0.03	0.55	0.38
1	1	84	14	54	0.06	-1.02	-0.78	-1.19	-1.11	-0.61	0.14
1	1	85	13	242	0.12	0.08	-0.88	0.41	0.06	-0.61	-0.43
1	1	86	9	373	0.25	1.38	0.92	0.03	0.06	0.55	-2.18
1	1	87	17	260	0.09	-0.62	0.92	0.31	0.06	-0.61	-0.70
1	1	88	12	282	0.3	-0.16	2.67	-0.10	-0.52	-0.61	-0.31
1	1	89	13	292	0.09	0.15	0.09	0.33	0.06	0.55	0.38
1	1	90	12	173	0.21	-0.82	0.09	-0.43	-0.75	0.55	-0.56
1	1	91	3	441	0.07	1.52	0.92	3.06	3.19	0.55	0.08
1	1	92	14	228	0.1	-0.32	-0.78	0.44	0.06	-0.61	1.52
1	1	93	22	124	0.08	-0.48	-0.78	-0.67	-0.72	-0.61	0.86
1	1	94	18	325	0.15	0.48	0.94	0.49	0.06	0.55	0.85
1	1	95	12	135	0.16	-1.06	0.09	-0.49	-0.72	-0.61	-0.66
1	1	96	12	87	0.06	-1.35	-0.78	-0.68	-0.72	-0.61	0.11
1	1	97	7	55	0.21	-0.74	0.45	-1.00	-0.94	0.55	-1.31
1	1	98	7	259	0.1	-0.55	-0.78	0.82	0.06	0.55	-0.88
1	1	99	5	204	0.09	-0.21	-0.78	-0.58	0.06	0.55	-0.99
1	1	100	14	212	0.16	-0.79	0.92	0.34	-0.72	-0.61	-0.53
1	1	101	11	88	0.08	-1.52	-0.78	-0.08	-0.72	-0.61	-0.68
1	1	102	12	196	0.16	0.71	-0.78	-0.16	-0.72	-0.61	0.50
1	1	103	8	269	0.22	-0.26	2.67	-0.73	-0.82	0.55	1.02
1	1	104	11	180	0.19	-0.57	0.92	-1.00	-0.97	0.55	0.15
1	1	105	9	354	0.23	-0.23	0.92	1.91	0.06	0.55	-0.95
1	1	106	20	402	0.26	0.86	0.92	0.71	1.62	0.55	1.24
1	1	107	14	122	0.08	-0.23	-0.78	-1.12	-0.72	-0.61	0.06
1	1	108	12	80	0.06	-1.49	-0.78	-0.64	-0.72	-0.61	0.49
1	1	109	9	97	0.06	-1.12	-0.78	-0.68	-0.72	-0.61	-0.01
1	1	110	14	287	0.11	-0.51	0.92	0.83	0.06	-0.61	-0.70
1	1	111	10	247	0.06	0.38	-0.78	0.33	0.06	-0.61	0.70
1	1	112	7	104	0.06	-0.34	-1.20	-1.02	-0.72	-0.61	0.48
1	1	113	10	218	0.09	-0.36	-0.78	0.43	0.06	-0.61	0.97
1	1	114	8	207	0.07	0.26	0.92	-0.68	-0.72	-0.61	0.64
1	1	115	7	348	0.1	1.55	0.92	0.83	0.06	-0.61	1.07
1	1	116	7	191	0.12	0.75	-0.78	-0.28	-0.72	-0.61	-0.06
1	1	117	16	69	0.14	-0.86	-0.83	-0.72	-0.72	-0.61	1.52
1	1	118	18	103	0.05	-0.88	-0.78	-0.66	-0.72	-0.61	0.90
1	1	119	12	24	0.12	-1.50	-1.16	-1.15	-0.78	-0.61	-0.46
1	1	120	6	34	0.12	0.41	-0.78	-0.77	-0.72	-0.61	-2.33
1	1	121	28	167	0.17	0.19	-0.89	-0.57	0.06	-0.61	-2.30
1	1	122	5	219	0.08	-0.85	-0.78	0.24	0.06	0.55	-0.33
1	1	123	6	288	0.16	0.22	-0.78	0.89	-0.07	0.55	0.48
1	1	124	24	395	0.13	1.20	0.92	0.45	1.62	0.55	0.43
1	1	125	14	369	0.14	1.26	0.09	0.69	1.62	-0.61	-0.08
1	1	126	8	23	0.06	-1.52	-1.20	-1.19	-1.11	-0.61	0.33
1	1	127	14	4	0.23	-1.28	-1.20	-0.96	-0.66	-0.61	-2.29
1	1	128	21	347	0.12	1.09	-0.80	0.68	1.62	-0.61	0.11
1	1	129	17	22	0.07	-0.39	-0.78	-1.01	-0.72	-0.61	-2.28
1	1	130	3	257	0.02	-0.45	-0.78	0.82	0.06	0.55	0.07
1	1	131	6	9	0.09	-0.91	0.09	-1.25	-1.11	-0.61	-2.06
1	1	132	20	277	0.11	0.46	0.92	-0.58	0.06	-0.61	-2.28
1	1	133	9	298	0.26	1.30	0.55	-0.70	-0.28	-0.61	-2.27
1	1	134	10	68	0.07	-1.55	-0.78	-0.63	-0.72	-0.61	0.87
1	1	135	9	208	0.06	-0.40	-1.20	0.33	0.06	-0.61	0.28
1	1	136	10	100	0.06	-0.67	-1.20	-0.65	-0.72	-0.61	-0.44
1	1	137	20	139	0.18	-0.65	-0.87	-0.62	-0.72	0.55	0.23
1	1	138	7	370	0.07	0.68	0.92	0.82	1.62	-0.61	0.44
1	1	139	12	95	0.05	-0.73	-0.78	-1.12	-0.72	-0.61	0.13
1	1	140	15	37	0.11	-1.42	-0.92	-0.69	-0.72	-0.61	1.52
1	1	141	8	263	0.17	0.50	0.09	-0.15	0.06	-0.61	1.03
1	1	142	11	295	0.11	0.73	0.09	-0.15	0.06	0.55	0.19
1	1	143	14	410	0.26	1.62	0.92	0.68	1.62	0.55	-0.25
1	1	144	6	3	0.22	-1.00	-0.92	-0.63	-0.91	2.88	-0.17
1	1	145	9	40	0.09	-0.56	1.38	-1.19	-1.11	-0.61	1.24
1	1	146	15	339	0.15	0.24	-1.03	0.82	1.62	-0.61	1.35
1	1	147	9	10	0.15	-0.74	-0.92	-0.76	-0.72	0.55	-2.35
1	1	148	20	187	0.16	-0.61	0.09	0.20	-0.72	-0.61	0.59
1	1	149	12	148	0.13	-1.07	0.92	-0.75	-0.72	-0.61	0.70
1	1	150	8	375	0.08	0.75	2.67	0.39	0.06	-0.61	1.52
1	1	151	14	389	0.21	2.47	0.92	0.72	0.06	0.55	0.32
1	1	152	18	344	0.09	0.62	-1.20	0.83	1.62	-0.61	0.74
1	1	153	7	309	0.1	0.30	0.92	0.82	0.06	-0.61	-0.69
1	1	154	17	357	0.32	0.84	-0.90	-0.13	1.62	-0.48	-2.27
1	1	155	10	332	0.18	0.12	0.96	0.75	0.06	0.55	1.52
1	1	156	16	129	0.17	-1.10	-0.81	-0.03	-0.72	-0.61	0.41
1	1	157	6	434	0.31	2.85	0.07	-0.06	1.62	2.88	-1.28
1	1	158	20	358	0.26	0.54	0.09	0.01	-0.05	2.88	0.53
1	1	159	13	179	0.28	0.42	-0.88	-0.67	-0.72	2.88	1.15
1	1	160	14	5	0.27	-0.61	-0.96	-0.68	-0.72	2.88	0.99
1	1	161	18	275	0.11	0.42	0.09	0.36	0.06	-0.61	0.73
1	1	162	12	8	0.08	-0.73	-1.20	-1.02	-0.72	-0.61	-2.33
1	1	163	14	310	0.12	1.00	0.92	0.15	0.06	-0.61	0.84
1	1	164	11	271	0.12	0.09	0.09	-0.16	0.06	0.55	0.68
1	1	165	5	250	0.11	-0.22	-0.95	0.33	0.06	0.55	-0.32
1	1	166	8	101	0.05	-0.78	-0.78	-0.89	-0.72	-0.61	0.78
1	1	167	9	426	0.31	2.13	2.67	1.77	0.06	-0.10	-0.57
1	1	168	16	156	0.11	-0.96	0.92	-0.82	-0.72	-0.61	0.08
1	1	169	4	98	0.07	-1.64	-0.78	-0.08	-0.72	-0.61	0.60
1	1	170	10	85	0.12	-1.26	-0.91	-0.71	-0.72	0.55	0.08
1	1	171	9	302	0.21	0.68	-0.01	-0.06	0.06	0.55	-0.58
1	1	172	10	351	0.1	0.96	-0.91	0.82	1.62	-0.61	-0.44
1	1	173	13	336	0.1	0.65	-1.13	0.62	1.62	-0.61	0.09
1	1	174	7	240	0.23	0.53	-0.78	-0.14	-0.49	0.55	0.81
1	1	175	6	329	0.14	1.40	0.92	0.58	0.06	-0.61	0.30
1	1	176	16	272	0.12	0.58	0.09	0.10	0.06	-0.61	0.16
1	1	177	10	429	0.06	1.03	-0.78	3.06	3.19	-0.61	-0.11
1	1	178	6	154	0.13	-1.33	-0.92	-0.08	0.06	-0.61	-0.92
1	1	179	14	366	0.13	0.40	2.67	0.51	0.06	0.55	0.44
1	1	180	7	15	0.17	-1.42	-0.96	-0.89	-0.83	0.55	1.52
1	1	181	10	153	0.15	-1.15	0.09	-0.21	-0.72	-0.61	0.41
1	1	182	2	236	0.06	-0.96	0.09	-0.08	0.06	0.55	0.08
1	1	183	14	131	0.23	-0.88	0.09	-0.91	-0.91	0.55	0.86
1	1	184	10	314	0.11	-0.47	0.92	0.86	0.06	0.55	-1.05
1	1	185	11	252	0.13	-0.16	0.09	0.30	0.06	-0.61	0.83
1	1	186	14	211	0.12	-0.43	-0.78	-0.27	0.06	0.55	0.78
1	1	187	6	185	0.16	0.21	0.78	-1.15	-0.72	-0.61	0.77
1	1	188	16	360	0.24	0.75	-0.86	0.62	1.62	0.55	-0.05
1	1	189	34	392	0.25	0.76	0.92	0.03	-0.01	2.88	0.51
1	1	190	11	400	0.28	1.26	0.77	0.08	1.62	0.55	-1.07
1	1	191	3	27	0.09	-1.97	-0.78	-0.50	-1.11	-0.61	-0.16
1	1	192	3	422	0.19	1.89	0.07	0.87	0.06	2.88	-1.08
1	1	193	20	186	0.14	0.01	0.09	-0.44	-0.72	-0.61	0.67
1	1	194	10	435	0.57	1.19	0.67	2.15	1.62	2.88	-0.66
1	1	195	4	401	0.04	1.09	1.38	0.82	1.62	-0.61	1.01
1	1	196	13	409	0.11	0.52	0.09	1.73	1.62	0.55	-0.93
1	1	197	9	385	0.26	0.83	-0.20	-0.45	-0.02	2.88	-1.23
1	1	198	4	308	0.09	2.01	-0.78	0.46	0.06	-0.61	0.52
1	1	199	5	166	0.24	-1.40	-0.95	0.10	0.06	-0.61	-0.08
1	1	200	15	294	0.2	0.12	0.92	-0.19	0.06	0.55	0.81
1	1	201	14	278	0.24	-0.52	0.92	0.60	0.01	-0.61	-1.30
1	1	202	9	81	0.07	-0.94	0.09	-1.19	-1.11	-0.61	0.41
1	1	203	2	379	0.07	0.22	1.15	-0.08	1.62	0.55	1.52
1	1	204	14	58	0.15	-0.41	-0.81	-0.73	0.06	-0.61	-2.32
1	1	205	7	449	0.71	3.86	1.92	1.68	0.29	2.88	-0.28
1	1	206	13	41	0.06	-1.48	-1.20	-1.14	-0.72	-0.61	0.50
1	1	207	7	73	0.06	-1.28	-0.78	-1.00	-0.72	-0.61	0.09
1	1	208	10	233	0.06	-0.40	-0.78	0.82	0.06	-0.61	0.07
1	1	209	5	19	0.1	-0.53	0.09	-1.14	-0.80	-0.61	-2.32
1	1	210	22	403	0.39	2.70	0.30	0.47	0.06	-0.61	-2.27
1	1	211	4	364	0.1	0.20	-0.99	0.82	1.62	0.55	1.27
1	1	212	4	324	0.08	1.11	0.92	0.12	0.06	-0.61	1.52
1	1	213	15	114	0.05	-0.88	-0.78	-0.65	-0.72	-0.61	0.10
1	1	214	13	155	0.1	-0.52	0.09	-0.73	-0.72	-0.61	0.26
1	1	215	7	172	0.1	-0.32	-0.84	-0.61	0.06	-0.61	0.87
1	1	216	8	416	0.41	0.96	-0.78	4.29	-0.03	-0.18	0.49
1	1	217	11	118	0.13	-0.91	0.92	-0.87	-0.86	-0.61	1.52
1	1	218	5	397	0.04	0.56	0.92	1.73	1.62	-0.61	0.07
1	1	219	11	318	0.13	0.94	0.92	-0.12	0.06	0.55	0.02
1	1	220	16	417	0.23	1.24	0.50	-0.55	-0.23	2.88	-2.30
1	1	221	9	39	0.06	-0.13	0.09	-1.01	-0.72	-0.61	-2.29
1	1	222	14	206	0.13	-0.50	-0.81	0.27	0.06	-0.61	-0.30
1	1	223	14	312	0.12	0.86	0.92	-0.36	0.06	0.55	0.57
1	1	224	11	136	0.08	-0.61	0.92	-1.19	-1.11	-0.61	0.27
1	1	225	4	29	0.06	-0.72	0.92	-1.19	-1.11	-0.61	-1.37
1	1	226	16	285	0.24	0.74	0.09	-0.23	-0.28	0.55	0.77
1	1	227	5	368	0.04	0.64	0.09	0.82	1.62	-0.61	1.52
1	1	228	7	376	0.17	0.69	-0.78	0.72	0.06	2.88	0.44
1	1	229	21	281	0.13	-0.25	0.92	0.59	0.06	-0.61	0.30
1	1	230	8	65	0.06	-1.29	-1.20	-0.66	-0.72	-0.61	0.70
1	1	231	10	431	0.06	1.19	-0.78	3.06	3.19	-0.61	0.47
1	1	232	5	411	0.04	1.15	1.38	0.82	1.62	-0.61	1.52
1	1	233	9	141	0.13	-0.90	0.92	-0.69	-0.72	-0.61	-1.12
1	1	234	10	353	0.06	0.27	-0.78	1.73	1.62	-0.61	0.07
1	1	235	13	443	0.16	1.49	0.09	3.06	3.19	0.55	-0.92
1	1	236	15	143	0.15	-0.41	-0.78	-0.72	-0.72	0.55	0.77
1	1	237	12	255	0.09	0.13	-0.78	0.85	0.06	-0.61	0.48
1	1	238	12	53	0.1	-0.51	-0.85	-0.78	-0.72	-0.61	-1.67
1	1	239	15	16	0.16	-1.19	-0.92	-1.04	-0.87	-0.61	-1.67
1	1	240	6	270	0.08	-0.05	0.09	0.41	0.06	-0.61	1.52
1	1	241	12	145	0.15	-0.03	-0.85	-0.65	-0.72	-0.61	0.60
1	1	242	12	35	0.06	-1.36	-0.78	-1.19	-1.11	-0.61	0.83
1	1	243	19	126	0.07	-0.55	-0.78	-0.65	-0.72	-0.61	0.10
1	1	244	11	327	0.16	0.68	0.92	-0.56	0.06	0.55	-2.10
1	1	245	8	363	0.16	0.73	2.67	0.52	0.06	-0.61	0.94
1	1	246	21	362	0.37	0.78	-0.90	0.36	1.62	-0.12	-1.36
1	1	247	12	42	0.13	-1.42	-0.82	-0.68	-0.72	-0.61	-1.28
1	1	248	4	14	0.09	-2.06	-0.78	-1.21	-1.11	-0.61	-0.19
1	1	249	19	202	0.08	0.14	0.92	-0.67	-0.72	-0.61	0.17
1	1	250	13	146	0.06	-0.38	0.92	-1.19	-1.11	-0.61	0.70
1	1	251	6	201	0.08	0.23	-0.78	-0.53	0.06	-0.61	0.25
1	1	252	8	33	0.05	-0.42	0.09	-0.89	-0.72	-0.61	-2.27
1	1	253	18	209	0.1	-0.48	-0.78	0.29	0.06	-0.61	0.43
1	1	254	12	323	0.06	0.19	-0.78	0.82	1.62	-0.61	0.44
1	1	255	8	222	0.17	0.91	0.09	-0.44	-0.72	-0.61	0.56
1	1	256	14	383	0.23	0.55	0.95	0.83	1.62	0.55	0.34
1	1	257	10	300	0.08	0.07	0.92	0.38	0.06	-0.61	1.52
1	1	258	5	51	0.07	-1.53	-0.78	-1.04	-0.72	-0.61	0.72
1	1	259	8	261	0.16	0.03	0.09	-0.35	0.06	0.55	0.10
1	1	260	14	138	0.06	-0.29	-0.78	-0.64	-0.72	-0.61	0.13
1	1	261	15	20	0.19	-0.78	-0.78	-0.05	-0.72	2.88	-0.12
1	1	262	16	342	0.09	0.68	-0.78	0.84	1.62	-0.61	0.82
1	1	263	7	158	0.08	0.08	-0.78	-0.58	-0.72	-0.61	-0.44
1	1	264	17	176	0.21	-0.46	0.09	-0.72	-0.79	0.55	0.31
1	1	265	13	447	0.59	1.10	-0.39	2.55	2.58	2.88	-1.02
1	1	266	10	200	0.12	-0.59	-1.20	0.43	0.06	-0.61	1.32
1	1	267	11	61	0.07	-1.02	0.09	-1.19	-1.11	-0.61	0.89
1	1	268	9	440	0.38	3.40	0.07	3.38	1.62	-0.61	0.74
1	1	269	10	404	0.06	0.35	-0.78	1.73	1.62	0.55	-1.30
1	1	270	7	195	0.15	-0.55	0.09	-0.51	0.06	-0.61	0.53
1	1	271	12	225	0.14	0.46	0.92	-0.58	-0.72	-0.61	-0.35
1	1	272	16	189	0.06	-0.08	0.92	-0.66	-0.72	-0.61	0.80
1	1	273	16	430	0.45	1.11	2.67	0.55	0.75	2.88	0.67
1	1	274	10	217	0.1	-0.05	-1.20	0.13	0.06	-0.61	0.10
1	1	275	2	164	0.07	-0.37	-0.78	0.34	-0.72	-0.61	-1.37
1	1	276	10	82	0.05	-0.62	-1.20	-1.12	-0.72	-0.61	0.17
1	1	277	8	47	0.15	-1.05	-0.83	-0.73	-0.72	0.55	-1.16
1	1	278	4	13	0.05	-2.09	-0.78	-1.21	-1.11	-0.61	0.76
1	1	279	19	159	0.19	0.12	-0.94	-0.80	-0.72	2.88	0.39
1	1	280	4	6	0.03	-0.76	0.92	-1.29	-1.11	-0.61	-2.25
1	1	281	14	160	0.14	-0.93	-0.78	-0.02	-0.72	0.55	0.48
1	1	282	11	170	0.29	-0.61	1.08	-0.96	-0.93	0.55	1.25
1	1	283	2	442	0.05	2.89	0.92	3.06	1.62	-0.61	-1.37
1	1	284	7	210	0.09	0.59	-0.78	-0.41	-0.72	0.55	-0.19
1	1	285	8	25	0.07	-1.55	-1.20	-1.16	-0.72	-0.61	0.90
1	1	286	15	279	0.18	0.42	0.92	-0.34	0.06	-0.61	-1.31
1	1	287	9	266	0.08	-0.33	0.09	0.82	0.06	-0.61	-0.74
1	1	288	6	79	0.14	-1.67	-0.78	-0.22	-0.85	0.55	0.07
1	1	289	6	276	0.22	-0.15	0.09	0.18	-0.20	0.55	1.52
1	1	290	11	28	0.1	-1.17	-0.78	-1.19	-1.11	0.55	0.66
1	1	291	15	102	0.1	-0.90	-0.78	-0.65	-0.72	-0.61	-0.63
1	1	292	5	161	0.08	-0.10	0.92	-0.85	-0.72	-0.61	-1.67
1	1	293	4	113	0.03	-0.29	-0.78	-1.14	-0.72	-0.61	0.87
1	1	294	17	341	0.15	0.38	2.67	0.54	0.06	-0.61	0.13
1	1	295	4	427	0.11	2.53	-0.78	-0.05	1.62	2.88	0.48
1	1	296	20	313	0.1	0.37	0.92	0.32	0.06	0.55	0.33
1	1	297	20	60	0.08	-1.45	-0.78	-0.67	-0.72	-0.61	-0.74
1	1	298	12	165	0.19	-0.22	0.09	-0.66	-0.65	-0.61	-0.93
1	1	299	13	245	0.09	0.28	-0.78	0.33	0.06	-0.61	0.13
1	1	300	12	157	0.17	-0.18	-0.92	-0.80	-0.72	0.55	0.42
1	1	301	2	444	0	2.90	0.92	0.32	4.75	0.55	0.50
1	1	302	5	229	0.16	-0.80	0.59	-0.08	0.06	-0.61	1.32
1	1	303	6	75	0.11	-0.72	0.09	-1.19	-1.11	-0.61	-0.65
1	1	304	7	119	0.11	0.29	0.09	-0.74	-0.72	-0.61	-2.32
1	1	305	13	26	0.06	-1.22	-1.20	-1.19	-1.11	-0.61	0.70
1	1	306	15	445	0.19	1.69	0.92	3.06	3.19	0.55	-0.91
1	1	307	22	289	0.27	0.51	0.09	-0.50	-0.19	0.55	-2.00
1	1	308	12	290	0.16	0.02	0.92	-0.27	0.06	0.55	0.14
1	1	309	11	340	0.17	1.29	0.92	0.18	0.06	0.55	0.95
1	1	310	6	184	0.13	-0.46	0.09	0.05	-0.72	-0.61	1.52
1	1	311	12	183	0.13	-0.22	1.26	-0.68	-0.72	-0.61	1.31
1	1	312	10	115	0.09	-0.46	-1.20	-0.71	-0.72	-0.61	0.63
1	1	313	10	256	0.15	0.42	0.09	-0.19	0.06	-0.61	-1.49
1	1	314	9	254	0.18	0.04	0.92	-0.41	0.06	-0.61	0.50
1	1	315	9	62	0.14	-0.97	0.09	-0.90	-0.89	-0.61	1.52
1	1	316	8	144	0.18	-0.16	-0.83	-0.61	-0.72	0.55	-1.33
1	1	317	4	418	0.13	2.77	0.92	0.48	0.06	0.55	-2.36
1	1	318	10	399	0.17	0.73	-0.78	-0.54	-0.09	2.88	-2.18
1	1	319	11	149	0.14	-1.20	0.92	-0.61	-0.72	-0.61	-0.60
1	1	320	12	36	0.07	-1.10	-1.20	-1.19	-1.11	-0.61	0.11
1	1	321	10	381	0.19	1.04	0.09	0.79	1.62	0.55	0.41
1	1	322	2	414	0.02	0.66	-0.78	3.06	1.62	-0.61	-0.30
1	1	323	4	241	0.08	-0.78	0.92	-0.08	0.06	-0.61	-0.46
1	1	324	8	66	0.15	-1.35	-0.78	-0.63	-0.77	0.55	-0.56
1	1	325	12	162	0.15	-1.18	-0.96	-0.08	0.06	-0.61	0.62
1	1	326	6	127	0.1	-0.66	-0.92	-0.65	-0.72	0.55	-0.45
1	1	327	15	77	0.07	-0.62	-0.78	-1.19	-1.11	-0.61	0.11
1	1	328	11	315	0.12	0.44	1.38	0.42	0.06	-0.61	1.29
1	1	329	12	346	0.31	2.27	0.64	0.55	-0.07	-0.61	0.36
1	1	330	13	44	0.07	-1.51	-1.20	-1.08	-0.72	-0.61	0.12
1	1	331	19	251	0.14	-0.18	-0.91	0.33	0.06	0.55	0.61
1	1	332	13	91	0.07	-0.83	-1.20	-0.71	-0.72	-0.61	0.70
1	1	333	8	230	0.06	-0.50	-0.78	0.82	0.06	-0.61	-1.30
1	1	334	17	226	0.07	-0.14	-0.78	0.33	0.06	-0.61	0.30
1	1	335	9	258	0.21	0.26	0.09	-0.57	-0.11	0.55	-1.08
1	1	336	18	356	0.35	-0.17	-0.11	3.06	0.06	-0.42	-0.81
1	1	337	3	321	0.07	0.84	0.92	0.83	0.06	-0.61	0.63
1	1	338	13	64	0.14	-1.52	-0.91	-0.65	-0.72	-0.61	-0.37
1	1	339	12	268	0.14	0.16	0.09	0.45	0.06	-0.61	0.35
1	1	340	12	365	0.23	0.41	2.67	0.44	0.00	0.55	-0.30
1	1	341	6	125	0.14	-0.90	-0.99	-0.85	0.06	-0.61	0.87
1	1	342	7	264	0.05	-0.35	-0.78	0.87	0.06	0.55	0.47
1	1	343	4	140	0.07	-1.53	0.09	-0.08	-0.72	-0.61	-0.19
1	1	344	15	175	0.14	0.01	-0.78	-0.08	-0.72	-0.61	0.76
1	1	345	17	56	0.12	-0.88	-0.86	-1.16	-0.95	-0.61	-0.44
1	1	346	18	110	0.13	-1.08	-0.78	-0.65	-0.72	0.55	0.68
1	1	347	12	205	0.1	-0.55	0.92	-0.60	-0.72	0.55	0.61
1	1	348	7	330	0.06	0.30	-1.20	0.82	1.62	-0.61	0.32
1	1	349	11	194	0.09	-0.47	-0.78	-0.53	0.06	0.55	0.11
1	1	350	10	38	0.06	-0.15	-0.78	-0.63	-0.72	-0.61	-2.33
1	1	351	9	108	0.11	-1.28	0.09	-0.90	-0.72	-0.61	-0.02
1	1	352	13	169	0.1	-0.12	0.92	-1.17	-1.02	-0.61	0.14
1	1	353	14	235	0.17	-0.23	0.98	0.25	-0.72	-0.61	0.59
1	1	354	18	296	0.12	-0.43	0.92	0.44	0.06	0.55	0.27
1	1	355	25	93	0.05	-1.20	-0.78	-0.68	-0.72	-0.61	0.48
1	1	356	10	306	0.3	-0.51	0.34	-0.16	-0.72	2.88	0.01
1	1	357	16	334	0.25	0.22	-0.78	-0.01	0.06	2.88	0.93
1	1	358	24	303	0.1	0.52	0.92	0.33	0.06	-0.61	0.55
1	1	359	5	265	0.05	-0.41	0.09	0.82	0.06	-0.61	-1.30
1	1	360	4	355	0.04	0.58	0.09	0.82	1.62	-0.61	1.01
1	1	361	14	243	0.16	-0.57	-0.84	0.32	0.06	0.55	1.45
1	1	362	8	291	0.07	0.02	0.09	0.34	0.06	0.55	0.93
1	1	363	9	17	0.06	-1.58	-1.20	-1.19	-1.11	-0.61	0.91
1	1	364	21	386	0.17	0.34	2.67	0.46	0.06	0.55	1.45
1	1	365	3	203	0.15	-0.99	0.64	-0.28	0.06	-0.61	-1.30
1	1	366	5	76	0.04	-0.96	-0.78	-1.14	-0.72	-0.61	0.89
1	1	367	23	352	0.11	1.09	-0.80	0.83	1.62	-0.61	0.68
1	1	368	10	168	0.11	-0.60	-0.82	0.36	-0.72	-0.61	0.51
1	1	369	21	408	0.14	1.65	0.92	0.68	1.62	0.55	0.66
1	1	370	2	198	0.1	1.06	-0.35	-0.55	-0.72	-0.61	-1.08
1	1	371	9	182	0.15	-0.12	0.92	-0.75	-0.72	-0.61	-1.05
1	1	372	16	96	0.11	-0.64	-0.83	-0.62	-0.72	-0.61	-1.15
1	1	373	24	132	0.23	-0.12	2.67	-0.87	-0.86	-0.61	1.13
1	1	374	9	253	0.15	0.48	0.92	-0.03	-0.72	-0.61	0.79
1	1	375	12	249	0.19	0.22	-0.85	-0.29	0.06	0.55	-1.47
1	1	376	13	299	0.16	-0.43	0.92	0.14	0.06	0.55	1.36
1	1	377	8	350	0.23	-0.16	0.61	-0.08	0.06	2.88	-0.14
1	1	378	18	18	0.08	-0.80	-0.78	-0.93	-0.72	-0.61	-2.30
1	1	379	19	216	0.07	-0.22	-1.20	0.33	0.06	-0.61	0.74
1	1	380	3	32	0.01	-0.46	-0.78	-0.50	-0.72	-0.61	-2.35
1	1	381	12	301	0.16	0.55	0.92	0.26	0.06	-0.61	-0.19
1	1	382	6	1	0.1	-1.47	-1.13	-1.29	-1.11	-0.61	-2.29
1	1	383	7	7	0.19	-1.24	-0.96	-1.20	-1.05	0.55	-1.33
1	1	384	9	197	0.19	-1.03	0.92	-0.33	-0.72	0.55	-0.56
1	1	385	6	311	0.11	0.48	0.92	0.84	0.06	-0.61	0.32
1	1	386	9	335	0.17	1.28	0.92	-0.50	-0.72	0.55	-2.24
1	1	387	17	439	0.1	1.20	-0.78	3.06	3.19	0.55	-1.11
1	1	388	7	283	0.14	-0.69	0.09	0.70	0.06	0.55	-0.87
1	1	389	20	359	0.31	0.64	-0.91	0.74	1.62	0.55	0.62
1	1	390	19	231	0.15	-0.52	-0.81	0.82	0.06	-0.61	-0.71
1	1	391	15	319	0.14	0.19	0.92	0.67	0.06	0.55	-0.41
1	1	392	14	59	0.07	-1.14	-1.20	-1.06	-0.72	-0.61	0.11
1	1	393	9	406	0.23	0.59	0.82	0.37	0.06	2.88	1.41
1	1	394	5	438	0.04	1.57	0.92	3.06	3.19	-0.61	0.47
1	1	395	6	405	0.17	2.16	0.92	0.73	1.62	-0.61	0.43
1	1	396	9	343	0.21	1.24	-0.78	-0.34	-0.72	2.88	-0.02
1	1	397	15	244	0.14	0.09	0.92	-0.58	-0.72	0.55	0.48
1	1	398	8	121	0.1	-1.10	-0.78	-0.03	-0.72	-0.61	0.98
1	1	399	15	150	0.08	-0.56	0.09	-0.71	-0.72	-0.61	0.83
1	1	400	7	46	0.05	-1.39	-1.20	-0.80	-0.72	-0.61	0.91
1	1	401	12	428	0.34	1.36	0.92	3.22	1.62	0.45	0.22
1	1	402	9	99	0.07	-0.83	0.92	-1.19	-1.11	-0.61	0.91
1	1	403	6	387	0.16	1.18	0.37	0.46	1.62	-0.61	-1.47
1	1	404	13	237	0.05	0.09	-0.78	0.33	0.06	-0.61	0.83
1	1	405	10	72	0.11	-0.69	-1.20	-1.14	-0.76	-0.61	0.61
1	1	406	7	43	0.1	-1.66	-1.20	-0.83	-0.72	-0.61	0.72
1	1	407	7	152	0.1	-0.80	-0.90	-0.66	0.06	-0.61	0.12
1	1	408	18	284	0.2	-0.66	0.92	0.21	-0.02	0.55	-0.48
1	1	409	11	45	0.05	-1.33	-0.78	-1.19	-1.11	-0.61	0.38
1	1	410	8	133	0.11	-0.16	0.09	-0.77	-0.72	-0.61	-1.67
1	1	411	10	70	0.06	-0.64	-0.78	-1.19	-1.11	-0.61	0.66
1	1	412	19	425	0.32	1.23	0.96	0.59	1.62	2.88	0.48
1	1	413	17	415	0.32	0.68	0.92	1.84	1.62	0.55	-0.97
1	1	414	12	171	0.08	-0.51	0.92	-0.73	-0.72	-0.61	0.90
1	1	415	7	92	0.05	-0.73	-0.78	-1.13	-0.72	-0.61	0.75
1	1	416	3	223	0.06	0.74	-0.78	-0.53	0.06	-0.61	0.30
1	1	417	12	328	0.11	0.23	-0.82	0.82	1.62	-0.61	-0.32
1	1	418	12	190	0.18	-0.85	0.09	-0.02	-0.72	0.55	0.48
1	1	419	20	128	0.05	-0.48	-0.78	-0.67	-0.72	-0.61	0.50
1	1	420	10	94	0.21	-1.18	-0.82	-0.08	-0.56	-0.61	1.52
1	1	421	6	412	0.38	0.64	1.21	3.24	0.06	-0.23	0.16
1	1	422	7	378	0.11	2.26	0.92	0.55	0.06	-0.61	-1.42
1	1	423	7	232	0.17	-0.06	0.92	-0.66	-0.72	0.55	-0.92
1	1	424	8	137	0.26	-1.51	-0.56	0.43	-0.72	0.55	-1.07
1	1	425	16	391	0.37	1.32	0.35	-0.36	-0.52	2.88	0.81
1	1	426	7	215	0.11	0.18	-0.90	-0.14	0.06	-0.61	-1.08
1	1	427	11	120	0.17	-0.44	0.09	-1.19	-1.07	-0.61	0.71
1	1	428	23	112	0.05	-0.85	-0.78	-0.69	-0.72	-0.61	0.43
1	1	429	6	107	0.1	-0.96	0.09	-0.71	-0.72	-0.61	-1.27
1	1	430	9	423	0.09	1.46	2.67	0.82	1.62	-0.61	1.29
1	1	431	11	31	0.16	-1.04	-0.90	-1.19	-1.11	0.55	-0.04
1	1	432	5	377	0.06	1.46	0.09	0.82	1.62	-0.61	0.35
1	1	433	9	274	0.13	-0.64	0.92	-0.03	0.06	0.55	0.33
1	1	434	7	446	0.8	2.12	0.92	5.82	0.40	-0.12	0.43
1	1	435	6	280	0.3	1.20	0.92	-0.31	-0.59	-0.61	-0.47
1	1	436	11	192	0.08	-0.44	-0.78	-0.08	0.06	-0.61	0.80
1	1	437	10	30	0.14	-0.26	0.92	-1.01	-0.76	-0.61	-2.30
1	1	438	7	398	0.15	1.43	0.92	0.71	1.62	-0.61	-0.62
1	1	439	5	130	0.1	0.16	-0.95	-1.14	-0.72	-0.61	0.55
1	1	440	11	221	0.14	-0.20	0.92	0.00	-0.72	-0.61	-0.13
1	1	441	11	227	0.07	0.27	-0.78	-0.08	0.06	-0.61	0.08
1	1	442	12	238	0.1	0.12	-0.82	0.30	0.06	-0.61	0.54
1	1	443	11	174	0.16	-0.30	0.92	-0.88	-0.86	-0.61	-0.51
1	1	444	15	178	0.11	-0.43	0.92	-0.72	-0.72	-0.61	0.24
1	1	445	8	177	0.1	-0.28	-0.78	-0.62	0.06	-0.61	-0.06
1	1	446	26	433	0.31	2.54	0.92	0.26	1.62	2.88	0.30
1	1	447	14	317	0.18	0.09	-0.78	-0.15	0.06	2.88	0.13
1	1	448	22	448	0.58	1.99	2.67	3.06	2.90	-0.30	0.21
1	1	449	14	116	0.06	-0.75	-0.78	-0.64	-0.72	-0.61	0.73
1	1	450	9	361	0.07	0.89	0.09	0.82	1.62	-0.61	0.46

Now let us understand what each column in the above table means:

Segment.Level - Level of the cell. In this case, we have performed Vector Quantization for depth 1. Hence Segment Level is 1.
Segment.Parent - Parent segment of the cell.
Segment.Child (Cell.Number) - The children of a particular cell. In this case, it is the total number of cells at which we achieved the defined compression percentage.
n - No of points in each cell.
Cell.ID - Cell_ID’s are generated for the multivariate data using 1-D Sammon’s Projection algorithm.
Quant.Error - Quantization Error for each cell.

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now, let’s check the compression summary for HVT (map A). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapA_compression_summary <- map_A[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapA_compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	450	371	0.82	n_cells: 450 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 82% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

hvt.results - A list containing the output of the HVT function which has the details of the tessellations to be plotted.
line.width - A vector indicating the line widths of the tessellation boundaries for each layer.
color.vec - A vector indicating the colors of the tessellations boundaries at each layer.
pch1 - Symbol type of the centroids of the tessellations (parent levels). Refer points (default = 21)
centroid.size - Size of centroids of first level tessellations (default = 3).
title - Set a title for the plot (default = NULL).

Let’s plot the Voronoi tessellation for layer 1 (map A).

muHVT::plotHVT(map_A,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1)

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 450 cells in the dataset ’computers’

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Let’s have a look at the function hvtHmap that we will use to overlay features as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

hvt.results - A list of results obtained from the HVT function (map_A).
dataset - A dataframe containing the variables to overlay as a heatmap. The user can pass an external dataset or the dataset that was used to perform hierarchical vector quantization. The dataset should have the same number of points as the dataset used to perform hierarchical Vector Quantization in the HVT function.
child.level - A number indicating the level for which the heat map is to be plotted.
hmap.cols - The column number of column name from the dataset indicating the variables for which the heat map is to be plotted. To plot the quantization error as heatmap, pass 'quant_error'. Similarly to plot the no of points in each cell as heatmap, pass 'no_of_points' as a parameter.
color.vec - A color vector such that length(color.vec) = child.level (default = NULL).
line.width - A line width vector such that length(line.width) = child.level (default = NULL).
palette.color - A number indicating the heat map color palette. 1 - rainbow, 2 - heat.colors, 3 - terrain.colors, 4 - topo.colors, 5 - cm.colors, 6 - BlCyGrYlRd (Blue,Cyan,Green,Yellow,Red) color (default = 6).
show.points - A boolean indicating whether the centroids should be plotted on the tessellations (default = FALSE).

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "n",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "price",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "speed",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "hd",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ram",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "screen",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ads",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 450,
)

Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

4 Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Flow map with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted map A and store it in identified_Novelty_cells variable.

The removeNovelty function removes the identified novelty cell(s) from the dataset and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) from the above table and the compressed HVT map (map A). It returns a list of two items: dataset with novelty records, and a subset of the dataset without the novelty records.

identified_Novelty_cells <<- c(77,203,231,262,318)
output_list <- removeNovelty(identified_Novelty_cells, map_A)

[1] “The following cell(s) have been removed as outliers from the dataset: 77 203 231 262 318”

data_with_novelty <- output_list[[1]]
dataset_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelties.For the sake of brevity, we will only show the first 10 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","price","speed","hd","ram","screen","ads")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = T, limit = 20)

Row.No	Cell.ID	Segment.Child	price	speed	hd	ram	screen	ads
1	450	77	5.2569137	0.9158993	4.0147755	4.746060	2.8768056	0.6272618
2	450	77	5.2569137	0.9158993	4.0147755	4.746060	2.8768056	0.2443755
3	450	77	5.2569137	0.9158993	4.0147755	4.746060	2.8768056	0.4996330
4	379	203	0.0471386	0.9158993	-0.0766612	1.623997	0.5489590	1.5206632
5	379	203	0.3832531	1.3793448	-0.0766612	1.623997	0.5489590	1.5206632
6	431	231	0.8807026	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
7	431	231	1.3815133	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
8	431	231	0.9798564	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
9	431	231	1.1327885	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
10	431	231	1.2319423	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
11	431	231	1.1294274	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
12	431	231	1.2991652	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
13	431	231	1.0403570	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
14	431	231	1.5512511	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
15	431	231	1.2924429	-0.7834010	3.0632786	3.185028	-0.6149643	0.4677258
16	342	262	0.8790220	-0.7834010	0.8748357	1.623997	-0.6149643	0.8665657
17	342	262	0.8807026	-0.7834010	0.8748357	1.623997	-0.6149643	0.8665657
18	342	262	0.7949934	-0.7834010	0.8748357	1.623997	-0.6149643	0.8665657
19	342	262	0.5445881	-0.7834010	0.8748357	1.623997	-0.6149643	0.8665657
20	342	262	0.7042425	-0.7834010	0.8748357	1.623997	-0.6149643	0.8665657

4.1 Voronoi Tessellation with highlighted novelty cell

The plotCells function is used to plot the Voronoi tessellation using the compressed HVT map (map A) and highlights the identified novelty cell(s) in red on the map.

plotCells(identified_Novelty_cells, map_A,line.width = c(0.4),centroid.size = 0.01)

Figure 11: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records to HVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

Number of Cells at each Level = 5
Maximum Depth = 1
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
dataset_with_novelty <- data_with_novelty[,-1:-2]
map_B <- list()
map_B <- muHVT::HVT(dataset_with_novelty,
                  n_cells = 5,
                  depth = 1,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F)

The datatable displayed below is the summary from map B (layer 2).

summaryTable(map_B[[3]]$summary,scroll = T,limit = 500)

Segment.Level	Segment.Parent	Segment.Child	n	Cell.ID	Quant.Error	price	speed	hd	ram	screen	ads
1	1	1	10	2	0.06	1.19	-0.78	3.06	3.19	-0.61	0.47
1	1	2	3	1	0.04	5.26	0.92	4.01	4.75	2.88	0.46
1	1	3	2	3	0.07	0.22	1.15	-0.08	1.62	0.55	1.52
1	1	4	16	4	0.09	0.68	-0.78	0.84	1.62	-0.61	0.82
1	1	5	10	5	0.17	0.73	-0.78	-0.54	-0.09	2.88	-2.18

5 Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 12:Flow map with highlighted bounding box in red around compressed map C

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty and below mentioned model parameters.

Model Parameters

Number of Cells at each Level = 10
Maximum Depth = 2
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

map_C <- list()
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F
                  )

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	10	0	0	n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2	100	2	0.02	n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 2% of the cells have hit the quantization threshold error in level 2

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters.

Model Parameters

Number of Cells at each Level = 23
Maximum Depth = 2
Quantization Error Threshold = 0.2
Error Metric = Max
Distance Metric = Manhattan

map_C <- list()
map_C <- muHVT::HVT(dataset_without_novelty,
                  n_cells = 23,    #23
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F
                  )

The datatable displayed below is the summary from map C (layer2).

summaryTable(map_C[[3]]$summary,scroll = T,limit = 500)

Segment.Level	Segment.Parent	Segment.Child	n	Cell.ID	Quant.Error	price	speed	hd	ram	screen	ads
1	1	1	381	311	0.56	-0.13	-0.86	0.32	-0.02	-0.42	0.44
1	1	2	104	173	0.58	1.83	0.78	0.51	-0.10	0.13	0.30
1	1	3	160	99	0.92	1.12	0.87	0.65	1.66	0.55	0.22
1	1	4	139	217	0.69	0.22	2.67	0.09	-0.21	-0.15	0.76
1	1	5	252	150	0.93	0.59	-0.85	0.99	1.60	-0.21	0.11
1	1	6	87	135	0.79	1.62	0.05	0.18	1.07	-0.33	-2.05
1	1	7	444	470	0.55	-1.30	-0.95	-0.83	-0.79	-0.61	0.67
1	1	8	150	27	1.49	1.54	0.04	3.29	2.63	0.26	-0.34
1	1	9	203	246	0.95	-0.37	0.48	0.97	0.04	-0.21	-0.79
1	1	10	269	448	0.57	-1.17	-0.77	-0.62	-0.72	-0.43	-0.82
1	1	11	267	395	0.49	-0.75	-0.68	-0.53	-0.62	0.55	0.47
1	1	12	286	393	0.59	-0.58	0.60	-0.76	-0.81	-0.52	0.82
1	1	13	104	233	0.71	0.81	0.40	-0.43	-0.15	1.02	-1.86
1	1	14	132	115	0.73	1.17	0.64	0.84	1.61	-0.61	0.52
1	1	15	155	355	0.67	0.17	0.47	-0.72	-0.35	-0.61	-2.03
1	1	16	287	240	0.51	0.19	0.70	0.10	-0.05	0.55	0.51
1	1	17	240	369	0.52	-0.48	0.67	-0.63	-0.75	-0.46	-0.29
1	1	18	375	420	0.38	-0.54	-0.87	-0.80	-0.71	-0.61	0.35
1	1	19	265	257	0.55	0.31	0.65	0.22	-0.07	-0.61	0.59
1	1	20	214	270	0.69	0.17	-0.26	-0.19	-0.35	2.88	0.36
1	1	21	150	54	1.11	1.76	0.96	0.58	0.84	2.88	0.21
1	1	22	78	42	0.87	1.50	2.67	1.66	1.78	-0.18	0.33
1	1	23	225	460	0.66	-0.39	-0.86	-0.77	-0.45	-0.40	-2.12
2	1	1	35	335	0.16	-0.34	-1.01	0.25	0.06	-0.61	0.87
2	1	2	18	302	0.13	-0.38	-0.85	0.82	0.06	-0.61	0.22
2	1	3	10	326	0.12	0.28	-0.78	-0.53	0.06	-0.61	0.33
2	1	4	34	298	0.1	0.12	-0.80	0.32	0.06	-0.61	0.76
2	1	5	6	274	0.14	0.97	-0.78	-0.08	0.06	-0.61	0.31
2	1	6	30	325	0.12	-0.44	-0.78	0.24	0.06	-0.61	0.41
2	1	7	24	297	0.1	0.12	-0.82	0.33	0.06	-0.61	0.38
2	1	8	2	266	0.11	1.01	-0.78	-0.08	-0.33	0.55	0.70
2	1	9	21	301	0.12	0.18	-0.82	0.10	0.06	-0.61	0.06
2	1	10	14	287	0.25	-0.01	-0.87	0.19	0.06	0.55	-0.12
2	1	11	9	310	0.17	-0.43	-0.87	0.44	0.06	0.55	1.41
2	1	12	13	360	0.14	0.05	-0.78	-0.08	-0.72	-0.61	0.80
2	1	13	14	331	0.13	0.70	-0.78	-0.08	-0.72	-0.61	0.26
2	1	14	8	316	0.15	-0.40	-0.94	0.83	0.06	-0.61	1.24
2	1	15	12	275	0.09	0.13	-0.78	0.85	0.06	-0.61	0.48
2	1	16	9	372	0.31	-1.15	-0.83	0.02	0.06	-0.61	0.43
2	1	17	18	322	0.12	-0.27	-1.20	0.26	0.06	-0.61	0.23
2	1	18	15	289	0.16	0.13	-0.84	0.49	0.06	-0.61	-0.47
2	1	19	16	340	0.12	-0.38	-0.91	0.33	0.06	-0.61	1.52
2	1	20	16	265	0.2	-0.16	-0.78	0.87	0.01	0.55	0.40
2	1	21	23	317	0.15	-0.50	-0.84	0.50	0.06	-0.61	-0.44
2	1	22	23	294	0.14	-0.19	-0.89	0.33	0.06	0.55	0.53
2	1	23	11	374	0.14	-0.60	-0.82	0.36	-0.72	-0.61	0.44
2	2	1	3	241	0.15	1.32	0.92	-0.23	-0.46	-0.61	-0.23
2	2	2	2	159	0.04	2.17	0.92	-0.02	0.06	0.55	-0.11
2	2	3	4	183	0.16	1.85	-0.13	0.45	0.06	0.55	0.26
2	2	4	6	179	0.12	1.36	0.92	0.37	0.06	0.55	0.86
2	2	5	6	186	0.18	2.25	0.37	0.46	0.06	-0.61	0.63
2	2	6	6	164	0.17	2.29	0.92	0.65	-0.20	-0.61	0.10
2	2	7	3	205	0.1	1.56	0.92	0.07	-0.72	0.55	-0.39
2	2	8	3	111	0.11	2.34	0.92	0.77	-0.20	0.55	-1.08
2	2	9	4	210	0.09	2.01	-0.78	0.46	0.06	-0.61	0.52
2	2	10	5	156	0.11	1.97	0.92	0.74	-0.72	0.55	0.60
2	2	11	5	165	0.2	1.83	0.92	0.70	0.06	-0.61	-0.99
2	2	12	9	181	0.13	1.51	0.92	0.71	0.06	-0.61	1.03
2	2	13	4	136	0.14	1.80	0.92	0.69	0.06	0.55	1.21
2	2	14	4	132	0.07	2.42	0.92	0.81	-0.72	0.55	0.10
2	2	15	4	184	0.1	1.54	0.92	0.71	0.06	-0.61	0.30
2	2	16	15	120	0.17	2.41	0.92	0.78	0.06	0.55	0.36
2	2	17	8	172	0.13	1.29	0.92	0.62	0.06	0.55	0.22
2	2	18	5	220	0.12	1.44	0.92	-0.32	-0.72	0.55	0.41
2	2	19	3	178	0.11	1.28	0.92	0.52	0.06	0.55	-0.44
2	2	20	5	202	0.07	1.24	0.92	-0.10	0.06	0.55	0.39
2	2	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	2	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	2	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	3	1	2	116	0.04	0.54	0.09	0.82	1.62	0.55	1.27
2	3	2	8	83	0.11	0.97	0.97	0.69	1.62	0.55	1.52
2	3	3	6	109	0.1	0.47	0.92	0.81	1.62	0.55	-0.96
2	3	4	4	80	0.06	1.73	0.92	0.65	1.62	0.55	-0.44
2	3	5	8	110	0.13	1.10	0.09	0.78	1.62	0.55	0.44
2	3	6	6	91	0.07	1.17	0.92	0.76	1.62	0.55	-0.37
2	3	7	9	97	0.11	0.77	0.97	0.82	1.62	0.55	1.01
2	3	8	12	123	0.23	0.39	0.99	0.67	1.62	0.55	0.17
2	3	9	4	74	0.12	2.15	0.92	0.51	1.62	0.55	0.17
2	3	10	4	84	0.06	1.58	0.92	0.83	1.62	0.55	0.14
2	3	11	2	87	0.1	0.96	0.09	1.27	1.62	0.55	-0.53
2	3	12	12	98	0.17	1.40	0.92	0.32	1.62	0.55	0.64
2	3	13	11	73	0.12	0.48	0.92	1.78	1.62	0.55	-1.09
2	3	14	2	11	0	2.90	0.92	0.32	4.75	0.55	0.50
2	3	15	4	95	0.1	1.43	0.92	0.07	1.62	0.55	-1.08
2	3	16	3	121	0.03	1.05	0.92	-0.08	1.62	0.55	-0.44
2	3	17	6	72	0.17	0.98	0.92	1.73	1.62	0.55	-0.50
2	3	18	9	108	0.08	1.45	0.92	-0.06	1.62	0.55	0.15
2	3	19	9	114	0.09	1.07	0.92	0.14	1.62	0.55	0.85
2	3	20	9	103	0.07	1.31	0.92	0.38	1.62	0.55	0.19
2	3	21	8	125	0.09	0.92	0.92	0.02	1.62	0.55	0.38
2	3	22	13	94	0.1	1.12	0.92	0.84	1.62	0.55	0.55
2	3	23	9	76	0.09	1.84	0.92	0.69	1.62	0.55	0.77
2	4	1	5	160	0.09	0.58	2.67	0.84	0.06	-0.61	0.68
2	4	2	3	232	0.12	-0.23	2.67	0.18	0.06	-0.61	-0.66
2	4	3	3	299	0.18	-0.19	2.67	-0.34	-0.72	0.55	0.08
2	4	4	8	157	0.08	0.75	2.67	0.39	0.06	-0.61	1.52
2	4	5	9	204	0.11	0.17	2.67	0.26	0.06	-0.61	1.52
2	4	6	25	158	0.23	0.43	2.67	0.48	0.06	0.55	0.11
2	4	7	2	492	0.01	0.00	2.67	-1.19	-1.11	-0.61	1.01
2	4	8	8	174	0.09	0.07	2.67	0.24	0.06	0.55	1.52
2	4	9	7	105	0.05	0.48	2.67	0.85	0.06	0.55	1.52
2	4	10	5	180	0.04	0.76	2.67	0.33	0.06	-0.61	1.01
2	4	11	5	379	0.04	0.14	2.67	-0.68	-0.72	-0.61	1.01
2	4	12	10	191	0.12	0.35	2.67	0.58	0.06	-0.61	-0.11
2	4	13	3	129	0.03	0.63	2.67	0.37	0.06	0.55	1.52
2	4	14	8	380	0.09	-0.21	2.67	-0.68	-0.72	-0.61	0.27
2	4	15	4	422	0.04	0.08	2.67	-0.68	-0.72	-0.61	1.52
2	4	16	4	511	0.04	-0.26	2.67	-1.19	-1.11	-0.61	1.52
2	4	17	3	169	0.07	0.45	2.67	0.19	0.06	0.55	1.01
2	4	18	6	208	0.13	0.31	2.67	0.26	0.06	-0.61	0.47
2	4	19	3	215	0.02	0.34	2.67	-0.08	0.06	-0.61	1.01
2	4	20	4	435	0.13	-0.57	2.67	-0.39	-0.72	-0.61	1.27
2	4	21	3	501	0.02	-0.33	2.67	-1.19	-1.11	-0.61	1.01
2	4	22	6	463	0.21	-0.28	2.67	-0.75	-0.85	0.55	1.27
2	4	23	5	284	0.25	-0.01	2.67	0.18	-0.72	-0.61	-0.41
2	5	1	5	130	0.03	0.43	-0.78	1.73	1.62	-0.61	0.07
2	5	2	12	82	0.09	0.49	0.09	1.73	1.62	0.55	-0.96
2	5	3	12	141	0.12	1.21	-0.82	0.73	1.62	-0.61	0.14
2	5	4	6	140	0.07	1.11	-0.85	0.82	1.62	-0.61	-0.44
2	5	5	7	89	0.13	0.43	-0.78	1.73	1.62	0.55	-0.42
2	5	6	13	143	0.27	0.89	-0.85	0.44	1.62	0.55	0.21
2	5	7	4	113	0.23	0.42	-0.78	3.32	0.06	-0.03	0.38
2	5	8	5	79	0.03	0.51	-0.78	1.73	1.62	0.55	-1.30
2	5	9	13	170	0.05	0.56	-0.78	0.82	1.62	-0.61	0.41
2	5	10	13	92	0.07	0.20	-0.78	1.73	1.62	0.55	-0.75
2	5	11	9	190	0.05	0.14	-0.78	0.82	1.62	-0.61	-0.30
2	5	12	6	175	0.1	0.68	-0.92	0.46	1.62	-0.61	-1.08
2	5	13	14	166	0.09	0.60	-1.20	0.69	1.62	-0.61	0.12
2	5	14	15	171	0.15	0.24	-1.03	0.82	1.62	-0.61	1.35
2	5	15	16	161	0.08	0.83	-0.78	0.66	1.62	-0.61	0.10
2	5	16	23	142	0.11	1.09	-0.80	0.83	1.62	-0.61	0.68
2	5	17	9	144	0.21	0.60	-0.87	0.60	1.62	0.55	-0.69
2	5	18	12	192	0.09	0.14	-0.89	0.82	1.62	-0.61	0.44
2	5	19	22	138	0.26	0.52	-0.93	0.83	1.62	0.55	0.71
2	5	20	5	145	0.03	0.11	-0.78	1.73	1.62	-0.61	0.07
2	5	21	7	162	0.08	0.63	-0.96	0.82	1.62	-0.61	-0.42
2	5	22	19	151	0.09	0.60	-1.20	0.83	1.62	-0.61	0.73
2	5	23	5	86	0.03	0.19	-0.78	1.73	1.62	0.55	-1.30
2	6	1	4	139	0.07	2.28	0.92	0.51	0.06	-0.61	-1.67
2	6	2	2	193	0.12	2.14	0.09	0.17	0.06	-0.61	-2.03
2	6	3	2	177	0.01	0.88	-0.78	-0.08	1.62	-0.61	-2.33
2	6	4	3	198	0.06	0.55	-0.78	-0.53	1.62	0.55	-2.04
2	6	5	1	147	0	1.05	-1.20	-0.08	1.62	-0.61	-2.39
2	6	6	3	187	0.02	0.71	-0.78	-0.08	1.62	-0.61	-2.28
2	6	7	4	106	0.06	1.30	0.92	0.46	1.62	-0.61	-1.37
2	6	8	2	119	0.03	1.55	-0.78	-0.08	1.62	-0.61	-2.39
2	6	9	8	90	0.26	1.28	0.71	-0.41	1.62	0.55	-2.09
2	6	10	4	134	0.06	1.13	0.09	0.46	1.62	-0.61	-1.37
2	6	11	4	188	0.03	0.58	-1.20	-0.08	1.62	-0.61	-2.25
2	6	12	4	152	0.03	0.88	-0.78	0.46	1.62	-0.61	-1.67
2	6	13	4	69	0.13	2.77	0.92	0.48	0.06	0.55	-2.36
2	6	14	4	107	0.03	1.26	0.92	-0.08	1.62	-0.61	-2.25
2	6	15	2	194	0.08	1.93	0.92	-0.33	0.06	-0.61	-2.38
2	6	16	4	146	0.03	1.09	0.09	-0.08	1.62	-0.61	-2.25
2	6	17	10	75	0.15	2.88	0.92	0.59	0.06	-0.61	-2.33
2	6	18	2	70	0.1	2.23	0.50	-0.08	1.62	-0.61	-2.39
2	6	19	6	117	0.23	1.07	-0.56	0.37	1.62	0.55	-1.47
2	6	20	2	163	0.01	0.62	-1.20	0.46	1.62	-0.61	-1.67
2	6	21	3	176	0.02	0.93	-0.78	-0.08	1.62	-0.61	-2.16
2	6	22	5	153	0.16	2.53	-0.78	0.46	0.06	-0.61	-2.18
2	6	23	4	102	0.08	2.86	0.09	0.46	0.06	-0.61	-2.30
2	7	1	21	507	0.1	-1.39	-1.20	-1.19	-1.11	-0.61	0.80
2	7	2	6	506	0.1	-1.17	-0.78	-1.16	-0.98	-0.61	1.52
2	7	3	36	466	0.1	-1.09	-1.20	-0.70	-0.72	-0.61	0.68
2	7	4	39	449	0.09	-1.05	-0.78	-0.71	-0.72	-0.61	0.90
2	7	5	23	424	0.22	-1.41	-0.86	-0.04	-0.62	-0.61	0.29
2	7	6	17	497	0.09	-1.27	-1.20	-1.19	-1.11	-0.61	0.22
2	7	7	16	495	0.1	-1.51	-1.14	-1.06	-0.72	-0.61	0.88
2	7	8	3	516	0.07	-1.65	-1.20	-1.19	-1.11	0.55	1.10
2	7	9	22	481	0.1	-1.31	-1.20	-1.06	-0.72	-0.61	0.11
2	7	10	10	455	0.21	-1.18	-0.82	-0.08	-0.56	-0.61	1.52
2	7	11	27	462	0.1	-1.36	-0.78	-0.84	-0.72	-0.61	0.08
2	7	12	16	480	0.24	-1.70	-1.12	-0.67	-0.77	-0.61	0.24
2	7	13	24	479	0.11	-1.14	-0.87	-0.68	-0.72	-0.61	1.52
2	7	14	47	447	0.22	-1.31	-0.76	-0.67	-0.72	-0.61	0.46
2	7	15	15	473	0.12	-1.57	-0.89	-0.65	-0.72	-0.61	0.88
2	7	16	23	491	0.11	-1.48	-1.16	-1.09	-0.72	-0.61	0.53
2	7	17	19	482	0.08	-1.18	-0.78	-1.19	-1.11	-0.61	0.34
2	7	18	9	510	0.13	-1.96	-0.83	-1.22	-1.11	-0.61	0.52
2	7	19	12	427	0.21	-1.22	-0.85	-0.04	-0.52	-0.61	0.94
2	7	20	27	488	0.09	-1.14	-0.78	-1.19	-1.11	-0.61	0.81
2	7	21	3	453	0.09	-1.06	-1.06	-1.02	0.06	-0.61	0.87
2	7	22	19	471	0.1	-1.05	-0.96	-1.14	-0.72	-0.61	0.80
2	7	23	10	515	0.18	-1.65	-1.15	-1.03	-0.87	-0.61	1.52
2	8	1	3	17	0.07	1.52	0.92	3.06	3.19	0.55	0.08
2	8	2	13	15	0.16	1.49	0.09	3.06	3.19	0.55	-0.92
2	8	3	2	1	0.07	2.26	0.92	8.30	1.62	0.55	0.27
2	8	4	5	20	0.04	1.57	0.92	3.06	3.19	-0.61	-0.30
2	8	5	10	30	0.06	1.19	-0.78	3.06	3.19	-0.61	0.07
2	8	6	6	10	0.07	2.13	0.92	3.06	3.19	0.55	-0.62
2	8	7	8	43	0.23	1.20	0.92	3.21	1.62	0.40	-0.33
2	8	8	9	12	0.38	3.40	0.07	3.38	1.62	-0.61	0.74
2	8	9	5	28	0.03	1.35	-0.78	3.06	3.19	-0.61	-0.30
2	8	10	6	3	0.29	1.11	-0.49	3.06	3.19	2.88	-1.07
2	8	11	5	31	0.03	1.03	-0.78	3.06	3.19	-0.61	-0.30
2	8	12	5	21	0.04	1.57	0.92	3.06	3.19	-0.61	0.47
2	8	13	2	18	0.05	2.89	0.92	3.06	1.62	-0.61	-1.37
2	8	14	4	25	0.05	1.17	-0.78	3.06	3.19	0.55	-0.11
2	8	15	4	6	0.33	1.66	0.92	3.25	2.40	2.88	-0.59
2	8	16	2	64	0.02	0.66	-0.78	3.06	1.62	-0.61	-0.30
2	8	17	2	24	0.02	1.17	-0.78	3.06	3.19	0.55	0.47
2	8	18	19	19	0.13	1.30	-0.78	3.06	3.19	0.55	-0.70
2	8	19	6	40	0.43	1.41	0.92	3.25	1.36	0.55	0.73
2	8	20	9	13	0.12	1.40	0.92	3.06	3.19	0.55	-1.10
2	8	21	10	36	0.53	1.65	0.07	4.82	-0.09	-0.38	0.49
2	8	22	10	14	0.06	1.28	-0.78	3.06	3.19	0.55	-1.30
2	8	23	5	23	0.04	1.49	0.92	3.06	3.19	-0.61	0.07
2	9	1	10	195	0.21	-0.15	-0.52	3.06	0.06	-0.61	-0.82
2	9	2	10	319	0.08	-0.66	-0.78	0.82	0.06	-0.61	-0.98
2	9	3	14	268	0.24	-0.77	0.92	0.21	-0.05	0.55	-0.64
2	9	4	7	292	0.12	-0.66	0.92	0.25	0.06	-0.61	-1.30
2	9	5	8	276	0.07	-0.38	0.09	0.82	0.06	-0.61	-0.76
2	9	6	4	223	0.09	0.69	0.09	0.74	0.06	-0.61	-1.02
2	9	7	17	280	0.09	-0.62	0.92	0.31	0.06	-0.61	-0.70
2	9	8	4	258	0.08	0.16	0.09	0.58	0.06	-0.61	-0.49
2	9	9	5	282	0.05	-0.41	0.09	0.82	0.06	-0.61	-1.30
2	9	10	7	254	0.06	-0.39	0.92	0.82	0.06	-0.61	-1.30
2	9	11	6	131	0.18	0.06	0.92	3.06	0.06	-0.61	-0.73
2	9	12	5	112	0.41	-0.25	0.42	3.14	0.06	0.55	-0.58
2	9	13	8	200	0.18	-0.29	0.92	1.78	0.06	0.40	0.07
2	9	14	8	189	0.11	-0.22	0.92	1.77	0.06	0.55	-0.96
2	9	15	8	303	0.13	-0.31	-0.78	0.77	0.06	-0.61	-1.16
2	9	16	8	219	0.14	0.34	0.92	0.82	0.06	-0.61	-0.77
2	9	17	13	225	0.16	-0.45	0.92	0.65	0.06	0.55	-1.20
2	9	18	19	261	0.13	-0.39	0.92	0.48	0.06	-0.61	-0.22
2	9	19	3	247	0.07	0.54	-0.78	0.84	0.06	-0.61	-1.23
2	9	20	16	281	0.36	-0.70	-0.29	0.78	-0.03	0.55	-0.90
2	9	21	3	327	0.09	-1.01	0.92	0.87	-0.72	-0.61	-0.85
2	9	22	8	211	0.09	-0.08	0.92	0.84	0.06	0.55	-0.55
2	9	23	12	256	0.09	-0.53	0.92	0.84	0.06	-0.61	-0.77
2	10	1	8	396	0.16	-1.33	-0.94	-0.08	0.06	-0.61	-0.76
2	10	2	10	496	0.12	-1.61	-1.20	-1.06	-0.72	-0.61	-0.43
2	10	3	8	454	0.28	-1.09	-0.46	-0.77	-0.77	0.55	-1.19
2	10	4	4	512	0.07	-1.39	-1.20	-1.04	-0.91	-0.61	-1.67
2	10	5	6	509	0.17	-1.28	-0.99	-1.21	-1.04	0.55	-1.28
2	10	6	6	500	0.08	-1.22	-0.78	-1.07	-0.91	-0.61	-1.67
2	10	7	27	457	0.15	-1.34	-0.80	-0.66	-0.72	-0.61	-0.59
2	10	8	16	398	0.25	-1.02	-0.78	0.29	-0.72	-0.61	-0.52
2	10	9	6	414	0.14	-1.29	-0.64	-0.08	-0.72	0.55	-0.62
2	10	10	7	401	0.14	-0.49	-0.84	-0.66	-0.49	0.55	-1.08
2	10	11	16	498	0.15	-1.31	-1.04	-1.11	-0.86	-0.61	-1.08
2	10	12	10	467	0.21	-1.28	-0.82	-0.75	-0.83	0.55	-0.61
2	10	13	17	438	0.09	-1.51	-0.78	-0.08	-0.72	-0.61	-0.54
2	10	14	18	429	0.14	-0.79	-0.92	-0.64	-0.72	-0.61	-0.48
2	10	15	14	416	0.21	-0.41	-0.81	-0.81	-0.49	-0.61	-1.05
2	10	16	9	485	0.11	-1.21	-0.97	-1.16	-0.98	-0.61	-0.44
2	10	17	13	472	0.17	-1.53	-0.82	-0.37	-0.72	-0.61	-1.30
2	10	18	12	407	0.3	-1.38	0.09	-0.32	-0.52	-0.61	-0.56
2	10	19	24	441	0.11	-0.84	-0.82	-0.69	-0.72	-0.61	-1.11
2	10	20	6	443	0.14	-1.58	-0.78	0.27	-0.72	0.55	-1.07
2	10	21	11	484	0.19	-1.80	-0.82	-0.75	-0.82	-0.61	-0.60
2	10	22	10	468	0.09	-0.81	-0.87	-1.15	-0.91	-0.61	-0.44
2	10	23	11	439	0.19	-0.98	0.09	-0.83	-0.82	-0.61	-1.19
2	11	1	8	493	0.16	-1.30	-0.89	-0.73	-0.77	0.55	1.52
2	11	2	11	383	0.14	-0.73	0.09	-0.61	-0.72	0.55	0.80
2	11	3	9	382	0.16	-0.74	-0.78	0.01	-0.72	0.55	0.60
2	11	4	11	359	0.11	-0.42	0.09	-0.64	-0.72	0.55	0.27
2	11	5	25	418	0.17	-0.77	-0.90	-0.67	-0.72	0.55	0.52
2	11	6	15	408	0.15	-0.58	-0.89	-0.65	-0.72	0.55	-0.08
2	11	7	13	324	0.12	-0.69	-0.78	0.08	0.06	0.55	0.41
2	11	8	7	330	0.09	0.59	-0.78	-0.41	-0.72	0.55	-0.19
2	11	9	4	333	0.1	0.33	-0.78	-0.08	-0.72	0.55	0.92
2	11	10	19	465	0.16	-1.21	-0.96	-0.75	-0.72	0.55	0.75
2	11	11	6	451	0.09	-0.92	0.09	-1.19	-1.11	0.55	0.69
2	11	12	15	338	0.18	-0.38	-0.78	-0.32	0.06	0.55	0.77
2	11	13	25	490	0.17	-1.16	-0.92	-1.18	-1.09	0.55	0.44
2	11	14	11	350	0.2	-0.99	0.09	-0.12	-0.58	0.55	-0.02
2	11	15	4	328	0.07	-0.91	-0.78	0.22	0.06	0.55	-0.33
2	11	16	11	344	0.18	-0.42	-0.78	-0.54	0.06	0.55	0.01
2	11	17	3	417	0.03	-0.58	0.09	-1.19	-1.11	0.55	0.12
2	11	18	7	445	0.15	-0.54	-0.90	-0.81	-0.72	0.55	1.30
2	11	19	7	342	0.15	-0.67	0.09	0.14	-0.72	0.55	0.80
2	11	20	20	394	0.16	-0.24	-0.85	-0.75	-0.72	0.55	0.60
2	11	21	7	343	0.12	-0.76	-0.78	0.15	0.06	0.55	1.37
2	11	22	14	402	0.17	-1.23	-0.78	0.01	-0.72	0.55	0.04
2	11	23	15	452	0.27	-1.31	-0.87	-0.70	-0.80	0.55	-0.02
2	12	1	16	363	0.25	-0.61	0.94	-0.76	-0.84	0.55	0.71
2	12	2	22	389	0.2	-0.58	0.09	-0.70	-0.54	-0.61	0.68
2	12	3	15	375	0.09	-0.44	0.92	-0.70	-0.72	-0.61	0.90
2	12	4	11	347	0.19	-0.70	0.92	-0.04	-0.65	-0.61	0.71
2	12	5	24	436	0.14	-0.64	0.09	-1.19	-1.11	-0.61	0.59
2	12	6	5	356	0.09	0.18	0.92	-1.15	-0.72	-0.61	0.75
2	12	7	5	464	0.04	-0.53	1.38	-1.19	-1.11	-0.61	1.01
2	12	8	23	346	0.08	0.05	0.92	-0.67	-0.72	-0.61	0.67
2	12	9	9	469	0.14	-0.97	0.09	-0.90	-0.89	-0.61	1.52
2	12	10	21	371	0.19	-0.07	0.09	-0.67	-0.72	-0.61	0.69
2	12	11	9	444	0.07	-0.83	0.92	-1.19	-1.11	-0.61	0.91
2	12	12	10	415	0.09	-0.60	0.92	-1.19	-1.11	-0.61	0.37
2	12	13	8	442	0.28	-0.82	0.99	-1.00	-0.96	0.55	1.39
2	12	14	11	406	0.13	-1.10	0.92	-0.76	-0.72	-0.61	0.72
2	12	15	10	364	0.14	-0.46	0.92	-0.72	-0.64	-0.61	0.45
2	12	16	18	373	0.19	-0.92	0.09	0.03	-0.67	-0.61	0.57
2	12	17	4	483	0.04	-0.93	0.92	-1.19	-1.11	-0.61	1.52
2	12	18	4	487	0.04	-0.59	1.38	-1.19	-1.11	-0.61	1.52
2	12	19	13	410	0.09	-0.33	0.92	-1.19	-1.11	-0.61	0.69
2	12	20	19	456	0.16	-1.19	0.09	-1.01	-0.90	-0.61	0.78
2	12	21	12	381	0.13	-0.22	1.26	-0.68	-0.72	-0.61	1.31
2	12	22	9	370	0.23	-0.68	0.09	0.09	-0.54	-0.61	1.35
2	12	23	8	413	0.22	-0.89	0.92	-0.61	-0.62	-0.61	1.52
2	13	1	6	293	0.11	0.40	-0.78	-0.22	0.06	0.55	-1.37
2	13	2	4	339	0.13	0.12	0.50	-0.68	-0.72	0.55	-1.37
2	13	3	2	255	0.08	1.11	0.92	-0.62	-0.72	0.55	-1.37
2	13	4	8	182	0.12	1.36	0.92	-0.08	0.06	0.55	-2.24
2	13	5	8	230	0.15	1.32	0.92	-0.48	-0.72	0.55	-2.31
2	13	6	1	126	0	1.55	0.92	0.87	0.06	0.55	-1.67
2	13	7	1	273	0	0.55	0.09	0.23	-0.72	0.55	-1.67
2	13	8	5	348	0.09	0.56	0.09	-0.59	-0.72	0.55	-2.34
2	13	9	3	308	0.01	0.75	-0.78	-0.50	0.06	0.55	-2.35
2	13	10	2	321	0.1	0.72	-0.35	-0.55	-0.72	0.55	-1.08
2	13	11	10	71	0.18	1.28	0.92	-0.54	-0.09	2.88	-2.18
2	13	12	11	224	0.15	0.64	0.92	-0.35	0.06	0.55	-1.29
2	13	13	7	288	0.08	0.20	0.09	-0.56	0.06	0.55	-1.08
2	13	14	8	228	0.12	0.72	0.92	-0.58	0.06	0.55	-2.27
2	13	15	3	227	0.11	0.65	0.09	0.10	0.06	0.55	-1.28
2	13	16	14	278	0.18	0.51	0.09	-0.53	0.06	0.55	-1.94
2	13	17	11	93	0.18	1.18	0.09	-0.54	-0.22	2.88	-2.13
2	13	18	0	NA	NA	NA	NA	NA	NA	NA	NA
2	13	19	0	NA	NA	NA	NA	NA	NA	NA	NA
2	13	20	0	NA	NA	NA	NA	NA	NA	NA	NA
2	13	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	13	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	13	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	1	10	148	0.11	0.71	0.09	0.82	1.62	-0.61	0.68
2	14	2	5	128	0.04	0.64	0.09	0.82	1.62	-0.61	1.52
2	14	3	14	127	0.11	1.31	0.09	0.69	1.62	-0.61	0.11
2	14	4	6	81	0.17	2.16	0.92	0.73	1.62	-0.61	0.43
2	14	5	4	124	0.06	1.21	0.09	0.82	1.62	-0.61	-0.44
2	14	6	15	88	0.15	1.01	1.19	0.82	1.62	-0.61	1.35
2	14	7	19	104	0.15	1.44	0.92	0.73	1.62	-0.61	-0.04
2	14	8	5	137	0.04	0.56	0.92	0.82	1.62	-0.61	-0.30
2	14	9	5	96	0.04	0.56	0.92	1.73	1.62	-0.61	0.07
2	14	10	21	100	0.1	1.42	0.92	0.83	1.62	-0.61	0.68
2	14	11	10	133	0.1	0.68	0.92	0.82	1.62	-0.61	0.61
2	14	12	17	122	0.08	1.27	0.09	0.83	1.62	-0.61	0.71
2	14	13	1	68	0	1.20	0.92	3.35	0.06	-0.61	0.50
2	14	14	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	15	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	16	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	17	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	18	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	19	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	20	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	14	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	15	1	19	426	0.13	-0.09	0.92	-1.03	-0.74	-0.61	-2.29
2	15	2	5	384	0.08	-0.10	0.92	-0.85	-0.72	-0.61	-1.67
2	15	3	5	354	0.18	0.41	0.09	-0.72	-0.72	-0.61	-1.20
2	15	4	5	475	0.1	-0.53	0.09	-1.14	-0.80	-0.61	-2.32
2	15	5	10	290	0.15	0.42	0.09	-0.19	0.06	-0.61	-1.49
2	15	6	7	405	0.11	0.29	0.09	-0.74	-0.72	-0.61	-2.32
2	15	7	4	504	0.03	-0.76	0.92	-1.29	-1.11	-0.61	-2.25
2	15	8	3	474	0.14	-0.69	0.92	-1.19	-1.11	-0.23	-1.67
2	15	9	5	357	0.14	1.15	0.59	-0.92	-0.72	-0.61	-2.32
2	15	10	8	312	0.06	0.19	0.92	-0.57	0.06	-0.61	-2.28
2	15	11	9	238	0.2	0.92	0.92	-0.29	0.06	-0.61	-1.34
2	15	12	7	409	0.06	-0.24	0.09	-0.76	-0.72	-0.61	-1.67
2	15	13	9	313	0.12	0.74	0.09	-0.64	0.06	-0.61	-2.31
2	15	14	20	341	0.09	0.17	0.09	-0.56	0.06	-0.61	-2.29
2	15	15	9	440	0.06	-0.13	0.09	-1.01	-0.72	-0.61	-2.29
2	15	16	2	345	0.14	-0.56	0.50	-0.60	0.06	-0.61	-1.19
2	15	17	13	277	0.14	0.70	0.92	-0.59	0.06	-0.61	-2.29
2	15	18	8	446	0.05	-0.42	0.09	-0.89	-0.72	-0.61	-2.27
2	15	19	7	291	0.15	0.15	0.92	-0.46	0.06	-0.61	-1.36
2	15	20	0	NA	NA	NA	NA	NA	NA	NA	NA
2	15	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	15	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	15	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	16	1	12	203	0.23	0.85	0.92	0.22	0.06	0.55	0.86
2	16	2	20	249	0.15	0.01	0.92	-0.28	0.06	0.55	0.42
2	16	3	8	296	0.2	0.56	0.09	-0.29	-0.72	0.55	0.71
2	16	4	10	251	0.15	0.17	0.09	0.08	0.06	0.55	0.03
2	16	5	8	306	0.14	-0.46	0.92	-0.03	-0.72	0.55	0.49
2	16	6	11	237	0.15	0.77	0.09	-0.22	0.06	0.55	-0.15
2	16	7	13	222	0.17	0.22	0.95	0.11	0.06	0.55	0.97
2	16	8	12	229	0.14	0.87	0.09	-0.14	0.06	0.55	0.78
2	16	9	16	252	0.2	-0.52	0.92	0.20	0.01	0.55	-0.09
2	16	10	7	314	0.12	0.15	0.92	-0.63	-0.72	0.55	0.90
2	16	11	10	267	0.11	0.11	0.09	-0.12	0.06	0.55	0.69
2	16	12	14	253	0.09	-0.49	0.92	0.18	0.06	0.55	0.47
2	16	13	14	207	0.19	0.49	0.92	0.46	0.06	0.55	-0.25
2	16	14	19	244	0.16	0.08	0.09	0.33	0.06	0.55	0.70
2	16	15	5	263	0.07	0.84	0.92	-0.60	-0.72	0.55	0.37
2	16	16	18	218	0.17	0.78	0.92	-0.32	0.06	0.55	0.37
2	16	17	17	214	0.1	-0.13	0.92	0.87	0.06	0.55	0.32
2	16	18	12	201	0.13	0.52	0.92	0.86	0.00	0.55	0.63
2	16	19	5	295	0.08	0.03	0.09	-0.62	0.06	0.55	0.30
2	16	20	11	309	0.12	0.00	0.92	-0.56	-0.72	0.55	0.30
2	16	21	14	260	0.22	-0.46	0.74	0.15	0.06	0.55	1.37
2	16	22	11	206	0.18	0.10	0.96	0.71	0.06	0.55	1.52
2	16	23	20	212	0.12	0.39	0.92	0.35	0.06	0.55	0.36
2	17	1	9	390	0.13	-0.67	0.09	-0.51	-0.72	-0.61	-0.58
2	17	2	18	352	0.14	0.17	0.09	-0.67	-0.72	-0.61	-0.02
2	17	3	20	332	0.13	0.10	0.92	-0.54	-0.72	-0.61	0.10
2	17	4	12	351	0.22	-0.99	0.85	-0.40	-0.72	0.55	-0.50
2	17	5	7	366	0.09	-0.49	0.92	-0.56	-0.72	-0.61	-0.69
2	17	6	4	432	0.06	-0.63	0.09	-1.19	-1.11	-0.61	-0.44
2	17	7	7	362	0.18	-0.95	0.09	-0.02	-0.49	-0.61	0.06
2	17	8	15	403	0.2	-1.13	0.92	-0.66	-0.72	-0.61	-0.88
2	17	9	7	388	0.21	-0.37	0.09	-0.66	-0.72	-0.28	-1.08
2	17	10	15	376	0.1	-0.64	0.92	-0.75	-0.72	-0.61	0.08
2	17	11	10	423	0.06	-0.48	0.09	-1.19	-1.11	-0.61	0.06
2	17	12	6	392	0.09	-0.56	0.09	-0.78	-0.72	-0.61	0.09
2	17	13	6	305	0.08	0.69	0.92	-0.56	-0.72	-0.61	-0.42
2	17	14	9	365	0.12	-0.13	0.92	-0.78	-0.72	-0.61	-1.10
2	17	15	7	307	0.16	-0.76	0.92	-0.14	0.06	-0.61	-0.24
2	17	16	18	358	0.29	-0.44	0.73	-0.83	-0.85	0.55	-0.35
2	17	17	20	337	0.18	-0.72	0.92	0.18	-0.72	-0.61	-0.31
2	17	18	11	399	0.18	-1.17	0.92	-0.79	-0.72	-0.61	0.00
2	17	19	7	411	0.15	-0.42	0.92	-1.18	-1.05	-0.61	-0.62
2	17	20	5	368	0.07	0.03	0.92	-1.15	-0.87	-0.61	0.14
2	17	21	11	329	0.2	0.06	0.92	-0.47	-0.65	-0.61	-0.48
2	17	22	7	431	0.07	-1.24	0.09	-0.96	-0.72	-0.61	0.06
2	17	23	9	397	0.05	-0.35	0.92	-1.19	-1.11	-0.61	0.06
2	18	1	66	421	0.14	-0.81	-0.78	-0.63	-0.72	-0.61	0.25
2	18	2	13	377	0.16	-0.58	-0.85	-0.64	0.06	-0.61	0.07
2	18	3	29	433	0.12	-0.70	-1.20	-0.69	-0.72	-0.61	0.26
2	18	4	31	450	0.13	-0.67	-0.80	-1.12	-0.84	-0.61	0.65
2	18	5	39	458	0.16	-0.74	-0.80	-1.16	-0.97	-0.61	0.08
2	18	6	9	387	0.09	-0.51	-0.87	-0.64	0.06	-0.61	0.87
2	18	7	11	367	0.18	0.49	-0.82	-0.59	-0.72	-0.61	0.27
2	18	8	22	419	0.17	-0.13	-0.84	-1.13	-0.74	-0.61	0.31
2	18	9	41	400	0.16	-0.29	-0.82	-0.64	-0.72	-0.61	-0.16
2	18	10	28	459	0.13	-0.63	-1.20	-1.09	-0.75	-0.61	0.35
2	18	11	39	404	0.14	-0.27	-0.84	-0.65	-0.72	-0.61	0.52
2	18	12	47	425	0.19	-0.60	-0.82	-0.66	-0.72	-0.61	0.85
2	18	13	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	14	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	15	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	16	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	17	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	18	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	19	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	20	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	21	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	22	0	NA	NA	NA	NA	NA	NA	NA	NA
2	18	23	0	NA	NA	NA	NA	NA	NA	NA	NA
2	19	1	9	283	0.25	0.46	0.09	-0.14	-0.02	-0.61	1.08
2	19	2	11	334	0.14	-0.07	0.09	0.10	-0.72	-0.61	0.58
2	19	3	8	243	0.22	0.18	0.92	0.34	-0.03	-0.61	1.52
2	19	4	9	264	0.08	0.59	0.09	-0.08	0.06	-0.61	0.18
2	19	5	7	242	0.11	0.28	0.92	0.27	0.06	-0.61	-0.38
2	19	6	5	286	0.12	0.18	0.92	-0.56	0.06	-0.61	0.29
2	19	7	12	231	0.23	0.96	0.92	-0.08	-0.07	-0.61	1.09
2	19	8	28	259	0.16	0.31	0.09	0.38	0.06	-0.61	0.64
2	19	9	6	285	0.18	0.78	0.92	-0.31	-0.72	-0.61	0.03
2	19	10	33	262	0.19	-0.31	0.92	0.38	0.06	-0.61	0.25
2	19	11	10	250	0.12	0.06	0.92	0.30	0.06	-0.61	0.96
2	19	12	12	245	0.12	0.51	0.09	0.37	0.06	-0.61	0.04
2	19	13	14	226	0.05	0.55	0.92	0.33	0.06	-0.61	0.74
2	19	14	14	221	0.17	0.88	0.92	0.05	0.06	-0.61	0.10
2	19	15	16	304	0.21	-0.07	0.97	0.23	-0.72	-0.61	0.59
2	19	16	11	216	0.12	0.44	1.38	0.42	0.06	-0.61	1.29
2	19	17	14	236	0.16	0.40	0.92	0.34	0.06	-0.61	0.26
2	19	18	8	315	0.17	0.91	0.09	-0.44	-0.72	-0.61	0.56
2	19	19	4	271	0.09	-0.22	0.92	0.23	0.06	-0.61	1.52
2	19	20	9	213	0.09	0.97	0.92	0.33	0.06	-0.61	0.69
2	19	21	6	300	0.1	-0.32	0.09	0.12	0.06	-0.61	0.55
2	19	22	11	279	0.12	-0.07	0.09	0.42	0.06	-0.61	1.29
2	19	23	8	209	0.08	0.66	0.92	0.84	0.06	-0.61	0.47
2	20	1	6	234	0.23	-0.18	-0.49	0.14	0.06	2.88	-0.42
2	20	2	8	235	0.17	1.26	-0.78	-0.37	-0.72	2.88	-0.12
2	20	3	16	185	0.25	0.64	0.09	-0.02	0.06	2.88	0.53
2	20	4	8	154	0.17	0.03	0.92	0.37	0.06	2.88	-0.48
2	20	5	15	478	0.2	0.48	-0.89	-0.81	-0.72	2.88	0.67
2	20	6	7	318	0.17	-0.47	0.92	-0.17	-0.72	2.88	-0.44
2	20	7	3	272	0.17	0.64	-0.78	-0.47	-0.20	2.88	-1.15
2	20	8	8	155	0.19	0.61	-0.78	0.74	0.06	2.88	0.39
2	20	9	6	486	0.21	-0.42	-0.85	0.12	-0.72	2.88	1.01
2	20	10	7	197	0.22	1.19	0.68	-0.71	-0.72	2.88	0.73
2	20	11	5	196	0.23	0.26	-0.61	0.35	0.06	2.88	1.42
2	20	12	9	239	0.11	0.27	-0.78	-0.19	0.06	2.88	0.12
2	20	13	7	269	0.24	-0.04	0.21	-0.08	-0.49	2.88	0.51
2	20	14	12	149	0.17	0.44	0.92	0.02	0.06	2.88	0.98
2	20	15	9	502	0.18	-0.84	-0.78	-0.15	-0.76	2.88	-0.60
2	20	16	8	199	0.24	1.17	-0.24	-0.08	-0.72	2.88	0.93
2	20	17	13	514	0.23	-0.28	-1.01	-0.87	-0.72	2.88	1.11
2	20	18	10	494	0.23	-0.89	-0.78	-0.16	-0.80	2.88	0.23
2	20	19	15	503	0.24	-0.37	-0.98	-0.74	-0.72	2.88	0.34
2	20	20	4	168	0.18	0.73	0.30	-0.40	0.06	2.88	-0.92
2	20	21	6	391	0.13	-0.63	0.09	-0.08	-0.72	2.88	-0.15
2	20	22	16	248	0.18	0.14	-0.78	-0.09	0.06	2.88	0.69
2	20	23	16	167	0.22	0.37	0.92	0.06	-0.08	2.88	0.30
2	21	1	4	46	0.14	0.87	2.67	0.49	0.06	2.88	1.52
2	21	2	9	65	0.26	2.46	0.92	0.02	-0.20	2.88	0.10
2	21	3	3	22	0.11	3.13	0.92	-0.06	1.62	2.88	-1.28
2	21	4	2	2	0.3	3.09	0.92	3.25	1.62	2.88	0.01
2	21	5	14	118	0.14	1.05	0.92	0.07	0.06	2.88	0.53
2	21	6	8	78	0.25	1.68	0.50	-0.08	0.06	2.88	1.06
2	21	7	8	38	0.19	0.85	-0.46	1.73	1.62	2.88	-0.93
2	21	8	3	32	0.12	2.57	-0.78	-0.06	1.62	2.88	-1.28
2	21	9	16	44	0.26	1.84	0.92	0.37	1.62	2.88	0.75
2	21	10	6	41	0.24	1.10	0.92	1.27	1.62	2.88	-0.83
2	21	11	8	85	0.2	0.85	0.92	0.87	0.06	2.88	0.63
2	21	12	4	39	0.11	2.53	-0.78	-0.05	1.62	2.88	0.48
2	21	13	3	58	0.19	1.89	0.07	0.87	0.06	2.88	-1.08
2	21	14	2	16	0.05	4.58	0.92	0.80	0.06	2.88	0.33
2	21	15	4	8	0.16	3.40	2.67	1.77	0.06	2.88	-0.54
2	21	16	9	33	0.23	2.59	0.92	0.70	1.62	2.88	0.20
2	21	17	10	34	0.22	2.78	0.92	-0.06	1.62	2.88	0.19

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)

segmentLevel	noOfCells	noOfCellsBelowQuantizationError	percentOfCellsBelowQuantizationErrorThreshold	parameters
1	23	0	0	n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2	494	418	0.85	n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 85% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

muHVT::plotHVT(map_C,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2)

Figure 13: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "n",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 14: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "price",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 15: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "speed",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 16: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "hd",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 17: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ram",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 18: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "screen",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 19: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ads",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
)

Figure 20: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

We now have the set of maps (map A, map B & map C) which will be used to predict which map and cell each test record is assigned to, but before that lets view our test dataset

6 Prediction on Test Data

Now once we have built the model, let us try to predict using our test dataset which cell and which layer each point belongs to.

The predictLayerHVT function is used to score the test data using the predictive set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Testing Dataset

The testing dataset includes the following columns:

Price: The price of 486 PCs in US dollars.
Speed: The clock speed of the PCs in MHz.
HD: The size of the hard drive in MB.
RAM: The size of RAM in MB.
Screen: The size of the screen in inches.
Ads: The number of listings for 486 PCs’ prices for each month.

Let’s have a look at our scaled test dataset containing 1253 datapoints

# Quick peek
testComputers <- scale(testComputers, center = scale_attr$`scaled:center`, scale = scale_attr$`scaled:scale`) %>% as.data.frame()
testComputers_data <- round(testComputers,4)
testComputers_data$Row.No <- row.names(testComputers_data)
testComputers_data <- testComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
Table(head(testComputers_data))

Row.No	price	speed	hd	ram	screen	ads
1	-1.2284	-0.7834	-0.6761	-0.7176	0.549	-0.8405
2	1.3832	0.0920	3.0633	3.1850	0.549	-0.8405
3	-0.8016	0.0920	-0.6761	-0.7176	-0.615	-0.8405
4	0.2303	2.6667	-0.4097	-0.7176	-0.615	-0.8405
5	0.3076	0.9159	1.7312	1.6240	0.549	-0.8405
6	-0.5075	0.9159	3.0633	0.0630	-0.615	-0.8405

Now let us check the structure of the test data and analyse its summary

str(testComputers)
#> 'data.frame':    1253 obs. of  6 variables:
#>  $ price : num  -1.228 1.383 -0.802 0.23 0.308 ...
#>  $ speed : num  -0.783 0.092 0.092 2.667 0.916 ...
#>  $ hd    : num  -0.676 3.063 -0.676 -0.41 1.731 ...
#>  $ ram   : num  -0.718 3.185 -0.718 -0.718 1.624 ...
#>  $ screen: num  0.549 0.549 -0.615 -0.615 0.549 ...
#>  $ ads   : num  -0.84 -0.84 -0.84 -0.84 -0.84 ...

summary(testComputers)
#>      price             speed               hd               ram          
#>  Min.   :-2.2216   Min.   :-0.7834   Min.   :-1.0995   Min.   :-1.10781  
#>  1st Qu.:-1.0368   1st Qu.: 0.0920   1st Qu.: 0.3420   1st Qu.: 0.06296  
#>  Median :-0.6167   Median : 0.9159   Median : 0.8986   Median : 0.06296  
#>  Mean   :-0.4276   Mean   : 0.9755   Mean   : 1.4372   Mean   : 0.60148  
#>  3rd Qu.: 0.1228   3rd Qu.: 1.3793   3rd Qu.: 2.3497   3rd Qu.: 1.62400  
#>  Max.   : 2.8957   Max.   : 2.6667   Max.   : 8.2965   Max.   : 4.74606  
#>      screen             ads         
#>  Min.   :-0.6150   Min.   :-3.2654  
#>  1st Qu.:-0.6150   1st Qu.:-1.8296  
#>  Median : 0.5490   Median :-1.4627  
#>  Mean   : 0.4672   Mean   :-1.7811  
#>  3rd Qu.: 0.5490   3rd Qu.:-1.2872  
#>  Max.   : 2.8768   Max.   : 0.7708

Now, Let us understand the predictLayerHVT function.

predictLayerHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = T, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of predictLayerHVT function has been explained below:

data - A dataframe containing the test dataset. The dataframe should have atleast one variable used for training. The variables from this dataset can also be used to overlay as heatmap.
map A - A list of hvt.results.model obtained from HVT function while performing hierarchical vector quantization on train data.
map B - A list of hvt.results.model obtained from HVT function while performing hierarchical vector quantization on train data with novelty.
map C - A list of hvt.results.model obtained from HVT function while performing hierarchical vector quantization on train data without novelty.
child.level - A number indicating the layer for which the heat map is to be plotted.(Only used if hmap.cols is not NULL).
mad.threshold - A numeric values indicating the permissible Mean Absolute Deviation.
normalize - A logical value indicating if the columns in your dataset should be normalized. Default value is TRUE.
distance_metric - The distance metric can be ’Euclidean” or “Manhattan”. Euclidean is selected by default.
error_metric - The error metric can be “mean” or “max”. mean is selected by default.
yVar - Name of the dependent variable(s).
... - color.vec and line.width can be passed from here.

The function predicts based on the HVT maps - map A, map B and map C, constructed using HVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Prediction Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the prediction method for a single point in the test dataset:

Calculate the distance between the point and the centroid of all the cells in the first level.
Find the cell whose centroid has minimum distance to the point.
Check if the cell drills down further to form more cells.
If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The prediction algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features

Let’s see which cell and layer each point belongs to.For the sake of brevity, we will only show the first 10 rows.


validation_data <- testComputers
new_predict <- predictLayerHVT(
    data=validation_data,
    map_A,
    map_B,
    map_C,
    normalize = F
  )

prediction_output <- head(new_predict,7)
row_indices <- c(343, 367, 423)
new_df <- new_predict[row_indices, ]
Predictions <- rbind(prediction_output,new_df)
row.names(Predictions) <- NULL

Predictions%>% head(100) %>%
  as.data.frame() %>%
  Table(scroll = T, limit = 20)

Row.Number	Layer1.Cell.ID	Layer2.Cell.ID
1	A66	C448
2	A443	C27
3	A107	C448
4	A282	C217
5	A415	C99
6	A356	C246
7	A443	C27
343	A399	B5
367	A399	B5
423	A399	B5

Note: From the above table, we can see that 343rd observation from the test data having Cell.ID as A399 have been identified as novelty and is mapped to 5th(B5) novelty cell in Layer2.Cell.ID column .Similarly, 367th,423rd……. test record having Cell.ID as A399 is correctly identified as novelty and gets mapped to B5 novelty cell in the Layer2.Cell.ID column respectively

muHVT: Predicting Cells with Layers using predictLayerHVT to monitor entities over time

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2023-06-20

1 Abstract

2 Example: muHVT with the Personal Computer dataset

3 Map A : Base Compressed Map

4 Map B : Compressed Novelty Map

4.1 Voronoi Tessellation with highlighted novelty cell

5 Map C : Compressed Map without Novelty

6 Prediction on Test Data

7 Executive Summary

8 References